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

Classics in Mathematics Kunihiko Kodaira

Complex Manifolds and Deformation of Complex Structures

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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

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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^*--«-

<'•'"'

where [ ] denotes the closure. Proof First we consider the case n = 2. In this case the right-hand side of (1.10) becomes ' ^'^ ''V(yi(^i),y2(^2))yU^i)y^(^2) , ^ ,^

\27Ti/ Jo Jo where the integrand is a continuous function of 6i and 62 for (zi, Z2) G Ur{c). Hence by the formula of the iterated integral, this integral is equal to

\2W

Jo y i ( ^ i ) - ^ i

Jo

y2(^2)-^2

Therefore by the Cauchy integral formula, the right-hand side of (1.10) becomes

yiTTi) J c i ^ i - ^ i

^ J Q ^2-^2

=JLf M l- ^ ^ ^ l = / ( ^ l , ^ 2 ) , 2m Jc, ^ 1 - ^1

which proves (1.10) in this case. Similarly for general n, by a repeated appHcation of the Cauchy integral formula, the right-hand side of (1.10) becomes

\27n7

Jc,^l-^l

\27r//

Jc„_i^n-l-'^n-l Jc„

J Ci ^ 1 - 2 : 1

Jc„_i

^n " ^n

^n-l-^n-1

_ _L r fUu ^2, . . . , ^n) •dCi=f{z^,.,,,z^). JTTI JJ c, c 2771

C\-Z\

As in the case of holomorphic functions of one variable, we shall deduce the fundamental properties of holomorphic functions of n variables from the integral formula (1.10).

4

1. Holomorphic Functions

First let (/^(^i,..., ^„) be a continuous function on C" = Ci x • • • x C„, and m i , . . . .rrin natural numbers. Consider the integral g(z) =

as a function of z = ( z , , . . . , z„) e Ur{c). Clearly giz) is continuous in UXc). Then for fixed Zze l/r2(ci),..., z„ e L/r^(c„), put ) df2 • • • dCn

<^(^i) is a continuous function of ^i on Q . Hence

f

g(z) = g ( z „ Z 2 , . . . , z „ ) = |

(^

71—^d^,

(1.12)

is a holomorphic function of Zi in (/nC^i)- Similarly g ( z i , . . . , z„) is a holomorphic function of each variable z^, /c= 1 , . . . , n, in Ur^{Ck). Hence g(z) = g ( z i , . . . , z„) is a holomorphic function of n variables z ^ , . . . , z„ in the polydisk Ur(c). By (1.12) we have d dZi

g ( z i , . . . , z j = mi

——

^^,dCx

Jc, (^i-^i) '

Thus (5g/5zi)(zi,..., z„) is also holomorphic in U^c). Similar results hold also for dg/dz^. By a repeated application of thisfesult to the right-hand side of (1.10), we obtain the following theorem. Theorem 1.2. A holomorphic function f{z) =f(zi,..., z„) ofn variables in a domain DczC is arbitrarily many times differentiable in Zi,..., z^ in D, and all its partial derivatives d'^''^""^'^''f{z)/dzT' * * * dz"^" are holomorphic in D. Moreover taking a polydisk U^c) such that [UXc)']^ D, we have d

*

"

dZi 1 • • • dZ„"

(27r0"

in Uric).

Jc"a.-z,)'"'"'---(^„-z„)"'""'

^^

^

§1.1. Holomorphic Functions

5

As in the case of functions of one variable we denote by Z'"'" •'""^(zi,..., z„) the partial derivative /(Zi,...,zJ dZi 1 • • • d Z „ "

of

f(z)=f(z^,...,Zn).

Theorem 1.3. Letf{z) =f(zi,...,Zn) be a holomorphic function in a domain D c= C", and c = ( c j , . . . , c„) G D. Then in a polydisk Up(^c) ^ D with centre c, / ( z ) has a power series expansion in Zi —c^,..., z„ — c„, Z

/(z)=

a„,..„„(z,-c,)'"'---(z„-c„)'"",

(1.15)

mi,...,m„=0

which is absolutely convergent in Up{c). The coefficient a^^...^^ is given by 1

-f-r-J(c,,...,c^).

(1.16)

Proof. By replacing z^ by z^ - c^, /c = 1 , . . . , n, we may assume that Ci = Cj = ••• = c„=0. For a point z = ( z i , . . . , Z J G (7^(0), p = ( p i , . . . , p j , take r = ( r i , . . . , r„) such that |zfc| < r^ < p^ for /c = 1 , . . . , /t. Then z 6 (7^(0), and [L/,(0)]c: (7^(0) c=D. Hence by (1.10) f(z)

\27ri) J C"

(1.17) (^l-^l) • • • (^n-^n)

Since |^fc| = rj^> |zfc|, /c = 1 , . . . , n, we have =i^
— = 7 2 (?)""•

fc=l,...,n.

Substituting these into the right-hand side of (1.17), we obtain a power series expansion /(z)=

Z

a„,..„„zr'---z-",

mi,...,m„=0

•dL

Letting M be the maximum of | / ( ^ i , . . . , ^„)| on C", we have M '1

'n

6

1. Holomorphic Functions

which proves that the above power series is absolutely convergent in Ur{0). From (1.14), it is clear that a^,...^ = / ' " ^ - '"«HO)/mi! • • • m j . I

(b) Power Series In this section we consider a power series oo

P(z) = P ( z „ . . . , z „ ) =

I

a„,...„„zr'• • • Z-"

mi,...,m„=0

with centre 0. If P{z) is a convergent at z, we denote its sum by the same notation P(z). Theorem 1.4. Let w = (Wi,..., w„) be such that W^T^O, ..., w„9^0. If P(z) is convergent at z = w, then P{z) is absolutely convergent for \zi\< | w i | , . . . , |z„| < |w„|, and its sum P(z) is a holomorphic function ofn variables z i , . . . , z„ in Up{0) where p = (|wi|,..., |>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<+oc. Hence M Pi'P2^ where Pi = | Wi| and p2 = | W2I. Therefore if |zi| < pi and \z^ < p2,

mi,m2 = 0

mi=0 \ P i /

m2 = 0 \ Pi /

namely, P(z) is absolutely convergent. Moreover taking arbitrary r^ and r2 with 0 < Ti < pi, 0 < r2 < P2, we have

|«mim2^1 ' ^ 2 l = | « m , m 2 k l ^ ^ 2 ^ ,

L |«m,m2Kl ^ ^2 ^ < + 0 0 mi,m2 —0

for z = (zi,Z2) with |zi|
H.l.

Holomorphic Functions

7

Replacing the variables z^ by z^ - c^, /c = 1 , . . . , n, we obtain a power series

Piz

-C) = PiZi

- Ci, . . . , Z„ - C„) =

I am,...mS^l m],...,m„=0

- C^)""' " ' {Zn " C^Y

with centre c = ( c i , . . . , c„). Corollaiy. If a power series P{z ~ c) is convergent at w = (wi,..., >v„) with Wi # C i , . . . , w„ 7^ c„, P(z — c) is absolutely convergent if |zfc — c^l < | w^ — c^l, fc = 1 , . . . , n, and its sum P{z — c) is a holomorphic function in Up{c), where p = (|wi-Ci), . . . , | w „ - c „ | ) . I The region of convergence of a power series P{z-c) is the union D = U Upic) of all polydisks Up(c) where P{z-c) is absolutely convergent. A region of convergence D is a domain if it is not empty. In case n = 1, the region of convergence of a power series is an empty set, an open disk, or the whole C itself, but in case n ^ 2, the region of convergence of a power series may take various forms. The next theorem follows immediately from this Corollary and Theorem 1.3. Theorem 1.5. A function f(z) = / ( z i , . . . , z„) ofn complex variables is holomorphic in a domain D c C" if and only iffor every point ce D, f{z) has a power series expansion P{z — c) which is convergent in some neighbourhood of c. I

(c) Cauchy-Riemann Equation First consider a continuously differentiable function / ( z ) of one complex variable z in a domain D c i C . Decompose z and / ( z ) into their real and imaginary parts by writing z = x + i_y a n d / ( z ) = w +/f. Then u and v are continuously differentiable functions of the real coordinates x, y in D. Using z and z, we have x = \{z^z),

>; = - : ( 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<^C. Since ^2fc-i — 2(^/c + ^/c)?

^2k — ~r:\^k~^k)'>

/c=l,...,n,

2i

we have

5z'fc

2\aX2fc-l

^^2fc/

^^/c

2\(9X2k-i

dX2k/

S i n c e / ( z ) = / ( z i , . . . , z„) is continuously differentiable, hence a fortiori continuous, / ( z ) is a holomorphic function of n variables z ^ , . . . , z„ if and only if/(z) is holomorphic in each z^ separately. Therefore from the above results, we obtain the following theorem. Theorem 1.6. Letf{z) = / ( z i , . . . , z„) he a continuously differentiable function ofn complex variables Zi,..., z^ina domain D c G". Thenf{z) is holomorphic in D if and only if ^

= 0,

fc=l,...,n.

I

(1.22)

§1.1. Holomorphic Functions

9

As is clear from Theorem 1.2, a holomorphic function/(z) = / ( z i , . . . , z„) in D c : C" is a C°° function in the real coordinates X i , . . . , X2„. A differential operator A= - r +

is called a Laplacian, and a C^-real function u = w ( x i , . . . , X2„) defined on a domain in IR^" is called a harmonic function if it satisfies the Laplace equation AM = 0.

Let/(z) = M + zi; be a holomorphic function of n complex variables Z i , . . . , z„. Then u and v are obviously harmonic functions since

—2 dX2k-l

1

i~ = 0, dX2k

—2 dX2k-l

'

~~^^ dX2k

/c=l,...,n.

Conversely in case n = l, let u = u(xi,X2) be a harmonic function in a domain D c i C . Then for any z e D , there is a holomorphic function/(z) defined in some neighbourhood of z such that u = R e / ( z ) there. This does not hold in case n ^ 2 . For example, in case n = 2, put r^ = Xi + - • • + X4. Then the function u = 1/r^ is harmonic, but d^u/dx^i + d^u/dxly^O, hence, u cannot be the real part of any holomorphic function. This already reveals the essential difference between holomorphic functions of one variable, and those of n variables with n ^ 2. (d) Analytic Continuation Theorem 1.7. Let f{x) =f(zi,..., z„) be a holomorphic function in D c : C". Unless f(z) is identically zero in D, at each point z of D, among f{z) and all its partial derivatives f^"^^''^''\z) at least one does not vanish. Proof Let DQ be the set of points zeD such that / ( z ) and all its partial derivatives vanish at z, and put D^- D- DQ. Then, D = DQU D^, D^n DQ = 0 and Do is open by Theorem 1.3. Since Di is clearly open, and D is connected, either D = DQ or D = Di holds. I Corollary. If two holomorphic functions f{z) and g{z) in a domain D coincide in some neighbourhood of a point ce D, thenf(z) and g{z) coincide in all of D, I

10

1. Holomorphic Functions

As in the case of holomorphic functions of one complex variable, this Corollary implies the uniqueness of the analytic continuation. First we must define the analytic continuation of a holomorphic function of n complex variables. Let Do be a domain in C", and fo{z) a holomorphic function in DQ. Let Di be another domain in C" with DQCSDI^^ 0 . If there is a holomorphic function/i(z) defined on D^ such that/i(z) =/o(z) on D o o D i , then /i(z) is said to be an analytic continuation of/o(z) to Dj. By the above Corollary, such/i(z) is unique if it exists. Let Di, D 2 , . . . , D „ , . . . be finitely or infinitely many domains in C", and /fc(z) a holomorphic function defined on each D^. If for each /c^ 1, /fc(z) is an analytic continuation of/fc_i(z), then every/^(z) is called an analytic continuation of/o(z). In this case putting/(z) =fk(z) for ze D^, we obtain a holomorphic function in D = DQU Dj u • • •. If this / ( z ) is one-valued in D, then / ( z ) is an analytic continuation of/o(z) to D in the above sense. In general, however, / ( z ) is not necessarily one-valued, but in these cases too, we call/(z) an analytic continuation of/o(z). As in the case of functions of one variable, by saying simply a holomorphic function, we mean a one-valued holomorphic function, whereas a holomorphic function which may not be one-valued is called an analytic function. Unlike in the case n = 1, in case n ^ 2 , there is a domain D Q C I C " for which there exists a domain D^DQ such that every holomorphic function defined on DQ can be continued analytically to D. To see this, first consider the case n = 2. Let pi, p2, CTI, 0*2 be real numbers with 0 < CTI < pi, 0 < 0-2 < P2, and let D be a polydisk with centre 0: D = Up(0) = {(zi, Z2) I |zi| < pi, IZ2I < P2},

P = (Pi, P2).

Put r(0) = {(zi, Z2) I Re zi ^ -0-1, IZ2I ^ cr2}, and Do= D— T{0). If we write Zj = Xj + 1X2, and Z2 = X3 + 1X4, the section of Do by the hyperplane X2 = 0 is illustrated in Fig. 1. Every holomorphic function defined on Do = D — T(0) has an analytic continuation to D. Proof. Let y/. 6 -> yXO) = (zi, re'^), 0 ^ ^ ^ 2-77, be the circle of radius r with centre ( Z I , 0 ) G D , where o-2
1-r

_| _ie do e z^

(1.23)

11

§1.1. Holomorphic Functions

•

^3

Figure 1

as a function of Zi and z^. Clearly /(z^, z^ is continuous in Dx = {(zi, Z2)G D | | z 2 | < r } , and holomorphic in each variable separately there. Hence/(zi, z^ is a holomorphic function of two variables Zi, Z2 in D^ Since for each fixed z with - p i < R e z < p i , (zi,Z2)eDo if |z2|
(1.24)

and Do= Up{c)- T{c), where z = ( z i , . . . , z„). Theorem 1.8 (Hartogs). Every holomorphic function /o(z) defined on the domain DQ= Up{c) — T{c) is continued analytically to a holomorphic function f{z) defined on Up(z).

12

1- Holomorphic Functions

Proof. Let y/. 0 -> yXO) = (z, re'^ + C2, Z 3 , . . . , z„), 0 ^ ^ ^ 277. By considering the integral fiiz)=z— 27n J

—

: i-Z2 l

rf^,

\Z2-C2\
instead of (1.23), the theorem is proved similarly as in the case above. I

n=2

Corollary. Let n^2. A holomorphic function fo(z) ofn variables defined on the punctured polydisk Up{x)-{c} is analytically continued to a holomorphic function f{z) defined on Up(c). I (e) Composite Function Let f{w) = / ( w i , . . . , w^) be a continuously differentiable function of m variables W i , . . . , w^ in a domain E c: C'", and let w, = gj(z) = g / ( z i , . . . , z„), 7 = 1,2, . . . , m , be continuously differentiable functions of n variables Z i , . . . , z „ in a domain Dc=C". Suppose g(z) = (gi(z),..., g„(z))e E if zeD. Let us consider the composite function/(g(z)) = / ( g i ( z ) , . . . , g„(z)). Clearly f(g(z)) is a continuously differentiable function of z = ( z ^ , . . . , z„) in D. The partial derivatives of/(z) with respect to z^ and 4 are given by

sf(g(z))^^/sfMs^_^sfM^^ j=i\ dwj azfc dwj dzj' dZk dfjgjz)) dZk

- (dfjw) dWj , dfjw) dwj\ = L\ -^—I -], jt,\

dWj

dZk

dWj

dzj'

' ^'^ Wj = gj(z). J

(1.26)

^J^ ^

Proof Decomposing w, and z^ into their real and imaginary parts by writing Wj = Uj-\-iVj, and Zfc = Xfc+%, and writing d/dZk, d/dzj,, and d/dWj, d/dWj as linear combinations of d/dx^, d/dy^ and d/dUj, d/dVj by (1.18), we obtain (1.25) and (1.26) from the usual chain rule for differentiation in the real variables. I Theorem 1.9. i f / ( w ) = / ( w i , . . . , w^) is holomorphic in Wi,..., w^, and Wj = gj(z) = gj{zi,..., z„), 7 = 1 , . . . , m, are holomorphic in Z i , . . . , z„, then the composite function f(g(z)) is a holomorphic function in Z j , . . , z„. Its partial derivatives are given by

^/(.(z))=i^^, oZk

j-1

dWj

dZk

». = , « .

(..27,

§1.1. Holomorphic Functions

13

Proof. Since by the assumption / ( w ) is holomorphic in Wi,..., w^, and Wj = gj{z) are holomorphic in Z i , . . . , z„, by Theorem 1.6, df{w)/dWj = Q, dwj/dzj, = 0. Therefore by (1.25) df{g{z))/dZk = 0 for A: = 1 , . . . , n. Hence by Theorem 1.6,/(g(z)) is a holomorphic function in z , , . . . , z„. (1.27) follows from (1.26). I (f) Weierstrass Preparation Theorem In this section we consider functions which are holomorphic in a neighbourhood of 0 = ( 0 , . . . , 0) 6 C^ For convenience sake we write (w, Z2,..., z„) instead of ( z i , . . . , z„). Let/( w, z) = / ( w, Z2,..., z„) be a holomorphic function defined in a neighbourhood of 0 = ( 0 , . . . , 0) such that /(0,0) = /(O, 0 , . . . , 0) = 0, and that /(w, 0) does not vanish identically. Let a i ( z ) , . . . , a^(z) be holomorphic functions in ( n - 1 ) variables ( z 2 , . . . , z„) such that ai(0) = • • • = ^^(0) = 0. Then the polynomial in w P(w, z) = w' + ai(z)>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„|<5. Since a;i(0) = • • • = w,(0) = 0, ai(0) = • • • = a,(0) = 0. Namely, P(w, z) is a distinguished polynomial. Put

P(w,z) Taking e > 0 and 6 > 0 sufficiently small, we may assume that /(w, z) is defined on |>v|<3e, | z 2 | < 6 , . . . , |z„|<6. Then, for any fixed z, u{w, z) is a holomorphic function of w in |w|<38. Hence we have

if M(W,Z) = -—

u{C,z) ^^ dC,

, , \w\<E.

(1.29)

Moreover M(W, z) does not vanish in | w | ^ e . On the other hand, since \a)x{z)\<e,.., ,\(x),{z)\<e, P{w,z)¥'Q for | w | ^ e . Hence M(W, z) is a holomorphic function of n variables w, Z2,..., z„ in e < |w| <3e, |z2| < 5 , . . . , |z„| < 5. Therefore by (1.29) M(W, Z) is a holomorphic function of n variables w, Z2,..., z„ in ^4,5(0). Moreover M( w, z) does not vanish there. I (g) Power Series Ring A power series Zmi,...,m„=o ^my-m^^T' * ' * ^^" whose region of convergence is not empty is called a convergent power series. The sum f=f(z) = Sm, m„=o ^mi • m„^r* ' ' * ^IT" of E convergcnt power series is a holomorphic

§1.1. Holomorphic Functions

15

function in a neighbourhood of 0. Conversely, a holomorphic function in a neighbourhood of 0 can be expanded into a convergent power series Xm„...,m„==o ^mr-mjT' * * * ^ ^ . ^Ui^) and giz) SLTQ holomorpMc in a neighbourhood of 0 , / ( z ) ± g ( z ) and/(z)g(z) are also holomorphic there. Therefore the set of all convergent power series forms a ring, which is called the power series ring and denoted by C { z i , . . . , z„}. I f / ( z ) and g{z) are holomorphic, and f{z)g{z) = 0 identically, then either / ( z ) = 0 or g(z) = 0 identically, hence the power series ring is an integral domain. In general an element u of an integral domain JR is called a unit, or an invertible element if there is an element v^R such that uv = 1 where 1 denotes the identity of R. Let f, ge R be non-zero elements. If there is an element he R such t h a t / = g/i, we say that g divides/ or t h a t / is divisible by g, and denote it by g\f. In this case g is called a divisor o f / and / is called a multiple of g. If g\f and/|g, / = ug with a unit u. In this case / and g are said to be associates. A non-zero element feRis called reducible if it can be written as a p r o d u c t / = gh where g, h are non-units. A non-zero element peR is called irreducible ifp is neither a unit nor a reducible element. If an element/e i? is written as a product of finite number of irreducible elements as f = Pl"'Pn,

(1.30)

/ i s said to bt factored into irreducible elements. (1.30) is called an irreducible factorization off We say th^tfhas the unique irreducible factorization up to units if the following condition is satisfied: L e t / = pi - • - Pm a n d / = qi ' ' ' qi be two irreducible factorizations of / Then we have / = m, and, after a suitable reordering, q^ and p^, are associates, that is, q^ = u^Pk with some unit Mfc, for /c = 1 , . . . , m. An integral domain R is called a unique factorization domain, or simply a UFD, if every non-zero element feR, which is not a unit, is factored into irreducible elements uniquely up to units. Let i^ be a UFD, and f = P\ • • • ;7^ an irreducible factorization o f / e R. Then p^, k = 1 , . . . , m, is called an irreducible factor off Non-zero elements / and g are said to be relatively prime or coprime i f / a n d g have no common irreducible factors. L e t / = /?! - - - Pm and g = ^i • • • ^/ be irreducible factorizations of/ and g. Then / and g are relatively prime if no pi and q^ are associates. This follows immediately from the fact that i^ is a UFD. Let R[w] be the polynomial ring in a variable w over an integral domain R. The following theorem is well known. Theorem 1.11. If R is a UFD, then R[w] is also a UFD. We denote the power series ring C { z i , . . . , z„} by OQ or OQI If/(z) G ^0 is a unit, /(O) # 0 obviously. Conversely, if/(O) T^ 0 in a small neighbourhood of 0, then l / / ( z ) is holomorphic, that is, l / / ( z ) e OQ. Thus / ( z ) G O'o is a unit if and only if /(O) T^ 0.

16

1. Holomorphic Functions

Now we prove that the power series ring €o = C{zi,..., z„} is a UFD. We proceed by induction on n. We write w instead of z, and/(w, z) instead of /(zi, Z2,..., z„) where z = ( z 2 , . . . , z„). In case n = 1, any non-zero element f{w)eC{w} which is not a unit can be written as / ( w ) = w\bs-^bs+iW-\-- • •) where s is a natural number, and b^T^O. Since w is irreducible and bs + bs+iW + ' • • is a unit, we see immediately that C{w} is a UFD. Now consider the case n>l.FutR = C{z2,..., z„}. Then R e R[w], and the set of the units of R[w] coincides with the set of the units of R. First consider an element/(w, z) e €Q such that/(0, 0) = 0 and that/(w, 0) 9^ 0. Then by the Weierstrass preparation theorem, /(w, z) can be written uniquely as f{w,z) = uP{w,z),

(1.31)

where M is a unit, and P(w, z) is a distinguished polynomial. Suppose that /(w, z) is written as a product m

/(>v,^)= n A ( w , ^ ) ,

(1.32)

where /fc(w, z) e O'o, and /^(O, 0) = 0, /c = 1 , . . . , m. Then, since n/c/fc(>^, 0) = /(w, 0) 7^ 0, we have /^(w, 0) 7^ 0, /c = 1 , . . . , m, hence each //c( w, z) can also be written as A(w,z) = WfeP,(w,z),

(1.33)

where u^ is a unit and P/c(>^> ^) is a distinguished polynomial. Then m

m

k=\

k=\

f{.w,z)= n «k n Pk{w,z). Since Hk w^ is a unit, and H^ ^fc(>v, z) is a distinguished polynomial, the uniqueness of the factorization (1.31) implies that P{w,z)=

n Pui^^.z).

(1.34)

fc=i

Thus the factorization (1.32) o f / ( w , z) = MP(W, z) corresponds to the factorization (1.34) of P(w, z) in JR[W]. Therefore if P{w, z) is irreducible in JR[w], it is also irreducible in €Q. In particular, putting n = 1 above, we see that P(w, z) is irreducible in ^0 if it is so in R{w\ Hence (1.32) is an irreducible factorization of/(w, z) in €Q if and only if (1.34) is an irreducible factorization of P(w, z) in P[w]. Therefore, in order to prove that €Q is a

§1.1. Holomorphic Functions

17

UFD, it suffices to show that any distinguished polynomial P{w, z) is factored uniquely into irreducible distinguished polynomials in R[w]. Since jR is a UFD by the hypothesis of induction, R[w] is also a UFD by Theorem 1.11. Put P(M;, z) = w' + ai(z)w'~^ + - • • + a,(z) with a^(0) = ' • • = a,(0) = 0, and let P{w,z)=U

a(w,z),

Qj,iw,z) = bUz)w''

+ "' + Ws,(z)

(1.35)

be the irreducible factorization of P(w, z) in R[w]. Then since 11^=1 ^fco(^) = 1, each bfco(^) is a unit in R. Consequently letting Pk(w,z) = bkoizy^Qk(yv,z) = w'^ + ak,(z)w''^~^-\-' • • + afc,^(z), we see that P/c(w, z) is also irreducible in P[w] and that P ( w , z ) = n i'i.Cw,^).

(1-36)

k=i

Since P(w, 0) = w\ Pki^^, 0) = w^^ Hence Pfc(w, z) is a distinguished polynomial. Since any unit in P[w] is actually a unit in P, Pfc(w, z) is the unique distinguished polynomial associated with Qk(y^^, z) in P[w]. Thus we have proved the uniqueness of irreducible factorizations up to units for/(w, z) e €Q such that /(O, 0) = 0 and that /(w, 0) T^ 0. A convergent power series / = / ( z i , . . . , z„) is said to be regular with respect to Zj if / ( z j , 0 , . . . , 0) T^ 0. Thus we have proved that \f feO^ is regular with respect to Zi, / is factored uniquely into irreducible elements up to units in €Q. Next suppose that / is not regular with respect to Zj. Introduce new coordinates z l , . . . , z^ by the linear transformation Zk=Y.Cjkz'j,

with

det(9fc)^0.

(1.37)

7=1

By this change of coordinates a convergent power series / ( z i , . . . , z„) in Z i , . . . , z„, becomes a convergent power series

/'(z;,...,^;)=/(zc,,zj,...,ic,„zj) in z ; , . . . , z ; . Hence ^o = C { z i , . . . , z„} = C { z ; , . . . , z;,}. We have / ( z j , 0 , . . . , 0) =f{cuz'u . . . , c^nz'i). Therefore unless / ( z i , . . . , z„) = 0 identically, choosing a suitable linear change (1.37), we have /'(z'l, 0 , . . . , 0) 7^ 0, that is, f'{z[,..., z^) is regular with respect to z\. Thus we obtain the following theorem: Theorem 1.12. The power series ring €Q = C { z i , . . . , z„} is a UFD.

I

18

1. Holomorphic Functions

Let /, g e OQ. Choosing, if necessary, a suitable linear change of coordinates, we assume t h a t / ( z i , . . . , z„) and g ( z i , . . . , z„) are regular with respect to Zi. We write w = Zi, and z = ( z 2 , . . . , z„) as before. Theorem 1.13. f=f{w, z) and g = g{w, z) are relatively prime in OQ if and only if there exist a(w, z), I3{w, z)eOo such that the equation a(w, z)/(w, z) + i8(w, z)g{w, z) = r{z),

r{z) ^ 0,

(1.38)

holds, where r(z)GC{z2,..., z„}. Here r(z)?^0 means that r(z) does not vanish identically. Proof If/(w, z) and g{w, z) have a common non-unit factor /z(w, z), then /z(w, 0) 7^ 0, and by (1.38) we obtain an equality q{w, z)h{w, z) = r{z),

q(w, z) e OQ.

From the proof of the Weierstrass preparation theorem given above, for every z = ( z 2 , . . . , z„) with | z 2 | < 6 , . . . , |z„|<5, the equation h{w,z) = 0 at least one solution w = (Oi{z). Consequently r{z) = has q(a)i{z), z)h{o)i{z),z) = 0 which is a contradiction. Thus the condition (1.38) is sufficient. Conversely, suppose that/(w, z) and g(w, z) are relatively prime. By the Weierstrass preparation theorem we can write /(w, z) = MP(W,

g{w, z) = vQ(w, z),

Z),

where w, v are units and P{w,z), Q{w, z) are distinguished polynomials. Then P{w, z) and (?(w, z) are relatively prime in i?[w], hence also in X[w] where K is the quotient field of R. Consequently using the Euclidean algorithm, we obtain an equation A{w)P{w, z)4-B(w)Q(w, z) = 1,

A{w), B{w) 6 X[w].

By multiplying a suitable non-zero element r{z)eR, r(z)B{w) e R[w], Putting a(w, z) = u~^r(z)A(w)

we obtain (1.38).

and

we have r(z)A(w),

/3(w, z) = M ~ V ( Z ) P ( W ) ,

I

The above results hold also for the power series ring C{zi - C i , . . . , z„ - c„} in Zi - C i , . . . , z„ - c„, which we denote by 0^, c = ( c i , . . . , c„). Let / ( z ) = / ( z i , . . . , z„) be a holomorphic function in Fez C". Then by Theorem 1.3, for every p e D,f{z) has a convergent power series expansion in Zi - C i , . . . , z„ - c„ in some neighbourhood of c: /(z) = P ( z - c ) = P ( z i - C i , . . . , z , - c J . We denote P(z-c)

by f{z).

§1.1. Holomorphic Functions

19

Theorem 1.14. Let f{z) and g{z) be holomorphic functions in D, Oe DaC". Suppose thatfoiz) and goiz) are relatively prime in OQ. Then if s>0 is small enough, fdz) and gdz) are relatively prime in Ocfor any c, |c| < e. Proof First suppose /(O) 9^ 0. Then for a sufficiently small e > 0, / ( c ) T^ 0 in |c| < s. Then fiz) is a unit in 0^, hence f(z) and gdz) are relatively prime. The case g(0)9^0 is proved similarly. Now suppose /(O) = 0 and g(0) = 0. Since fo{z) and go{z) are relatively prime in OQ, choosing coordinates z , , . . . , z„ such that /o(z) and go{z) are regular with respect to Zj, we see from Theorem 1.13 above that there exist a(z), p{z)e OQ such that a(z)fo{z) + p(z)go{z) =

r{z)^0.

Since for a sufficiently small s>0, a(z), f3{z) are holomorphic in | z | < e , we have aiz)f(z)

+ I3{z)g(z) = r ( z 2 , . . . , z J 7^ 0,

\z\ < e.

Hence if |c|< e adz)fc{z) + lidz)gdz)

= rdz2....,

z j ^ 0,

where r^Czs,..., z„) is the power series expansion of r ( z 2 , . . . , z„) with centre (c2, . . . , c „ ) . Hence Theorem 1.13 proves t h a t / ( z ) and gXz) are relatively prime in ^o- I Definition 1.2. Let/(z) and g{z) be holomorphic in a domain D c C " . / ( z ) and g{z^ are said to be relatively prime in D if fiz) and gdz) are relatively prime in 0^ at every point ce D. By Theorem 1.14 above, if fdz) and gdz) are relatively prime in Oc at a point ceD, f(z) and g{z) are relatively prime in a sufficiently small neighbourhood U{c) of c. (h) Analytic Hypersurface Let / ( z ) be a holomorphic function in a domain DciC". The set 5 = {ze D\f(z) = 0} is called an analytic hypersurface. Let 0 G 5 and choose coordinates z^, Z2,..., z„ such that/(z) = / ( z i , . . . , z„) is regular with respect to Zi. We write w = Zj, and z = ( z 2 , . . . , z„) as before. Then by the Weierstrass preparation theorem, fiw,z) is represented in a polydisk Ue,s(0) = {(w, z) 11 w| < e, IZ2I < 6 , . . . , |zj < 5}, as f(w,z)

= uP{w,z),

P{w,z) = w' + a,(z)w'-'

+ -" + adz),

20

1. Holomorphic Functions

where u does not vanish in U^siO), and P(w, z) is a distinguished polynomial. Consequently S is given in U^^sW by the following algebraic equation of degree s in w: w' + ai(z)w'"' + - • • + fl,(z) = 0. If we denote the roots of this equation by w i ( z ) , . . . , ^^(z), then S n U,,s{0) = {ia)^(z), z)\k=l,2,...,

s,\z2\< 8,...

,\z^\<8}.

a>i(z),..., (JL>s{z) are continuous functions of z = ( z 2 , . . . , z„), which is in general multi-valued if 5 ^ 2. Note that a^{0) = - - - = a^iO) = 0 implies that wi(0) = • . • = co,{0) = 0. For each z e Us{0) = {z \ \z2\ < 6 , . . . , |z„| < 5} the set {(Ok{z)\k=l,.., ,s} consists of at most k points. Thus S n t/^ 5(0) is a hypersurface which is the orbit of these sets as z moves on the polydisk 1/5(0). The following theorem is an analogy of the Riemann extension theorem for holomorphic functions of one complex variable. Theorem 1.15. Let S be an analytic hypersurface in a domain DciC". Any bounded holomorphic function / i ( z i , . . . , z„) defined onD-S can be extended to a holomorphic function h(zi,..., z^) in D. Proof We may assume that Oe S and [U^s(0)]cz D. We write w = Zi, and z = ( z 2 , . . . , z„) as before. It suffices to show that h{w, z) can be continued analytically to a holomorphic function g(w, z) in U^^sW- For any fixed z e C/e,s(0), h{w, z) is a function of w which is holomorphic in |w| < e except at a ) i ( z ) , . . . , w^(z), and by the assumption h{w,z) is bounded. Hence /i(w, z) is extended to a holomorphic function g{w, z) of w in |w|< e. Let C be the circle of radius e: |w| = E in the w-plane. By the Cauchy integral formula we obtain

I f g{^.z)=—-\ ^TTlJc

h{C-z) ^^ — d^, (-Z

Hence g(w, z) is a holomorphic function of n variables w, Z2,..., z„ in t/e,.(0).

I

Let S = {fiw, z) = 0} as before. S is called irreducible at 0 if the power series expansion/o(w, z) of/( w, z) at 0 is irreducible in €Q = C{W, Z2,..., z„}. In this case the equation /o( w, z) = 0 is called an irreducible equation of S at 0. As stated in (g) above, the irreducibility of/o(w, z) in ^0 is equivalent tothatof P(w, z)in jR[w]. Inthecase5= l,P(>v, z) = w + ai(z) is irreducible, and ft>i(z) = - a i ( z ) is holomorphic in L^sCO).

§1.1. Holomorphic Functions

21

Suppose 5 ^ 2 . If P(w, z) is irreducible, P(w, z) and P^(w,z) = sw'~^ ^ {s - l)ai{z)w'~^-h' ' ' + as-i{z) are relatively prime in R[w]. Consequently by Theorem 1.13 above there exist a(w, z), p(w, z) e R[w] such that a(w,

Z)P(M;, Z) + /3(W, Z)P^(W,

Z)

= riz) 9^ 0,

r^z) e R,

(1.39)

where r(z) = r ( z 2 , . . . , z„) is a holomorphic function in Us(0) with r(0) = 0, and r(z) 7^0 means that r(z) does not vanish identically on Us{0). Thus A = {zG (75(0)|r(z) = 0} is an analytic hypersurface in Us{0). If a;y(z) is a multiple root of the equation P(w, z) = 0, PH;(<W;(Z), Z) = 0, hence by (1.39), r{z) = 0. Therefore for z e Us{0)-A, the 5 roots a;i(z),..., ^^^(z) are all distinct: (Oj(z) 9^ (Oj^{z) (j T^ k). Hence coi{z),..., (Os(z) are possibly multivalued holomorphic functions of z in L^5(0)-A. If z moves along a closed curve in Us(0) — A, (Oi{z) is continued analytically to one of aj'^(z)'s. Thus any analytic continuation of a>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 = <^(z) of a domain D c C " into C". Using complex coordinates, we can write O as follows: O : Z = ( Z i , . . . , Z j -^ ( W i , ...,Wm)

= {(Pliz),

. . . ,

(Pm{z)).

24

1. Holomorphic Functions

is called continuous, continuously differentiable C°° in D if (pi{z),..., (pm{z) are continuous, continuously differentiable, C°° in L>, respectively. Definition 1.3.0 is said to be holomorphic in D if Wj = : z ^ w = 0 ( z ) is holomorphic, the matrix

w^


dWi

azi \^^k/7=l,...,m

dWrn dZi

dZn

is called the Jacohian matrix of , and denoted by a ( w i , . . . , w ^ ) / a ( z , , . . . , z„). In particular if m = «, the determinant of the Jacobian matrix of O: / ( z ) = det

a(>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 = <|)(z) be a holomorphic map of D c C " into C", and ^ : w ^ ^ = ^ ( w ) a holomorphic map of E c z C " into C". If 0 ( D ) c : £, the composite '^ ° 0 : z -> f = '^((z)) is a holomorphic map of D into C*'. The Jacobian matrix of '^ ° is the product of those of ^ and <E>:

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'=:
Then g, are holomorphic functions of vvi,..., w^, z^+i,..., z„ in (7(w^), and rank — = v. a ( w i , . . . , w^, z ^ + i , . . . , z j

§1.2. Holomorphic Map

27

Therefore, since gi = W i , . . . , g^ = w^, we have dg^+j

. ^ d(gu

• . . , g^, gv+j)

r»

•

1

7

1

Hence g^+^(>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<^ M into N. Since Zji p-^ Zj(p) maps Uj homeomorphically onto %, and W;,: q-^Wxiq) maps W;^ homeomorphically onto W;^, for A,7 such that

^,j:zj(p)^wAq),

q = ^{p),

pe^-\W,)nUj,

is a continuous map from the domain %;, ={Zj(p)\pe(^~\W;,)n into W^:

(2.5) L^jc C"

cE>

^-\W,)nUj %.

^W,

b

> ^ .

If ;^^ is of class C or holomorphic, is said to be of class C^ or holomorphic where r = 1, 2 , . . . , 00. If there exists a biholomorphic map w\ p->w{p) for a domain W<^ M into C , we can use this w as local complex coordinates. Thus if M " is covered by at most countably many domains W j , . . . , W),,..., and for each W), a biholomorphic map W;^: /?-> 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 D<^ M into N is written as O:

zj^w^=^{zj).

Thus we can dispense with the indices 7, A in O;^^ where w^, = Ox^(z^). (c) Locally Finite Coverings Let S be a Hausdorff space. An open covering U of X is said to be locally finite if for each point pel., there is a neighbourhood U{p) such that U(p) n U 9^0 for only a finite number of the members L'^e U. Let M " be a complex manifold, and q a point of M. By local coordinates with centre q we mean a biholomorphic map Zq\ p-^ Zq{p) of a domain U{q) containing q onto a domain of C" containing ( 0 , . . . , 0) such that Zq{q) = 0. Let Zq'. p ^ Zq{p) = {z\{p),..., z''q{p)) bc givcn local coordinates with centre q. By a coordinate polydisk UXq) with centre q on M", we mean the inverse image

uM)=^~,\Urm={p\Wq{p)\
§2.1. Complex Manifolds

33

by Zq of the polydisk with centre 0 ^.(0) = {z|z = ( z ^ . . . , z J , | z ^ | < r ^ . . . , | z " | < r " } , such that [(7,(0)] c:z^((7(^)), where r = {r\ ... .r"") and r ' > O f o r all /. Theorem 2.1. Let M"" he a complex manifold. Suppose that for each point qe M, local coordinates z^ with centre q and a coordinate polydisk UjK^q^iq) are given. Then we can choose at most countably many coordinate polydisks Uj = Uru)(qj)^ Uj,^q^(qj),

j

=1,2,3,...,

such that L^ = { L^ |j = 1, 2, 3,...} is a locally finite open covering of M. Proof Omitted.

I

(d) Submanifolds Let M be a complex manifold. In the sequel we fix for each point qe M local coordinates Zqi p-> Zq{p) with centre q arbitrarily. Definition 2.4. Let S" be a closed subset of M". S is called an analytic subset of M if for each qe S, there are a finite number of holomorphic functions flip),' • ">fq(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<^q)(q) with respect to the local coordinates z^: p^ ^qip) such that UR(^q)(q) n S = 0 for q?^ S and that L^R(g) ^ U{q) for qe S. Then by Theorem 2.1, we may choose coordinate polydisks Uj= UruMj)"^ UR^q.^(qj),

7 = 1,2,...,

such that U = { L7,17 = 1, 2,...} is a locally finite open covering. For simplicity we write Zj for Zq.,fj{p) for fg.{p) and Vj for i^iqj). Then either S n Uj = 0, or qj e S, and if qj e S, SnU

= {peUj\fj(p)

= "-=fp(p)

= 0},

(2.8)

In particular if 5 is a complex submanifold, we have SnUj = {peUj\zr\p)

= "' = zJ(p) = Ol

(2.9)

where m = n-Pj is independent of 7. In fact if 5 n L^ n L4 7^ 0 , it is clear that rrij = m^. Then the assertion follows from the connectedness of S. From this result, we see that S is itself a complex manifold. For, if we put Vj = Sn Uj, V = {Vj\ Vj 9^ 0} is 3, locally finite open covering of S, and zjs:p^zjsip)

=

(z](p),...,zr{p)))

is a homeomorphism of Vj onto a polydisk in C". The coordinate transformation TjkS'

Zksip)-^Zjs{p)

§2.1. Complex Manifolds

35

is biholomorphic since it is the restriction of TJ^: /^l

^^

-,"\ ^ / ^ l

^"1

(Zfc, . . . , Z f c , . . . , Z f c ) ^ ( z ^ - , . . . , Z ^ -

,...,Zj)

-r"\

+^

f^^'+l

to

{Zfc

-"

Al

= . . . = Zfc = 0 } .

Consequently {Zjs | VJ T^ 0 } forms a system of local complex coordinates, hence, defines a complex structure on S. Thus 5 is a complex manifold of dimension m. An analytic subset 5 of M " is called an analytic hypersurface if it is defined by a single equation {fqip) = 0} in some neighbourhood of each qe S. In this case, choosing Uj as in (2.8), we have SnUj = {peUj\fj(p) = 0},

(2.10)

whcTQ fj(p) is holomorphic in Uj. If we identify Uj with % = Zj{Uj)<^C" as usual, Sn% = {zjemj\fjizj) = 0} is an analytic hypersurface of %, where fj(p)=fj{Zj) is a holomorphic function of Zj in %. Then if we take U^i^q^iq) sufficiently small, by the Corollary to Theorem 1.18, we may choose ^(z,) such that^(z^) = 0 is a minimal equation of 5 n % in %. In this case fj(p) = 0 is called a minimal equation of S in Uj. Then S is smooth at ^ G 5 n U, if and only if at least one of the partial derivatives dfj(Zj)/dzf offj(Zj) does not vanish at Zj = Zj(q). Consequently the set 5 ' of all smooth points of S is an open subset of S. Let S"= S-S' be the set of singular points of S. Then 5"'n % is defined by the system of holomorphic equations

Thus 5" is an analytic subset of M". M" — S" is a complex manifold, and each connected component of S' = S-S" is a submanifold of M" - S " . (e) Meromorphic Functions Let M " be a complex manifold. For each point g G M",fixlocal coordinates Zq-. p^(z]j,..., Zq) = Zq(p) with centre q. A holomorphic function h{p) defined in a domain Da M" is expanded into a convergent power series /Zq(Zg) == hq(z]j,..., Zq) of z ^ , . . . , Zg in a neighbourhood of each point ^G D. Namely we have h{ p) = hq{Zq(p)) ina. neighbourhood of q. Suppose given holomorphic functions h{p) and g(p) in D. h{p) and g(p) are called relatively prime at q if their power series expansions hq{zq) and gqizq) are relatively prime in C { z ^ , . . . , Zq}. hq(Zq) and gq{Zq) have a common divisor

36

2. Complex Manifolds

in C { z ^ , . . . , z^} if and only if there are holomorphic functions d, h, and gi defined in a small neighbourhood of q such that h(p) = h,(p)d(p),

gip) = gi(p)d(p),

and

^(^) = 0.

(2.11)

Consequently the above definition does not depend on the choice of local coordinates z^. Since C j z ^ , . . . , z^} is a UFD, we may choose d(p) in (2.11) such that hi and gi are relatively prime at q. This being the case, we call d(p) the greatest common divisor of h{p) and g{p) at q. We say that h{p) and g(/?) are relatively prime in D if they are so at every point of D. By Theorem 1.14, if h and g are relatively prime at a point qeD, they are relatively prime in a sufficiently small neighbourhood U{q)^ D of q. A function f{p) on M " is called a meromorphic function if for each point qeM"", there are holomorphic functions hq{p) and gq(/?) defined in some neighbourhood of q such thdii f{p) = hq{p)/gq{p) there. From the above argument we may assume that hq{p) and gq{p) are relatively prime at q, hence, also in some neighbourhood U{q) of q. Consequently by Theorem 2.1, as in (d) above, there is a locally finite open covering U = { L^ ly = 1, 2,...} such that each U, is a coordinate polydisk, and that on each Uj,f(p) is represented as the quotient of two relatively prime holomorphic functions fj and gj: f(p)=^,

for peU,

(2.12)

mP) For each pair 7, k with Ujn U^T^ 0 , we have hj(p)gk(p) = h(p)gj(p)

for peUjn

U^.

(2.13)

Since hjq(Zq) and gjq(zq) are relatively prime, and also h^qizq) and gkqi^q) are relatively prime, hjq(Zq) and h^qizq) are associates, and gjq(zq) and gkq(zq) are also associates. Therefore

Kip)

gk(p)

is a non-vanishing holomorphic function defined in Uj n Uk, and we have hj(p) =fjk(p)hk(p)

and

gj(p) = fjkip)gkip),

peUjn

U^.

LQtf(p) = hq{p)/gq(p) be a meromorphic function, and suppose gq(q) = 0. In case n = 1, since hq(p) and gq(p) are relatively prime, gqiq) = 0 impHes that hq(q)9^0, hence / ( ; ' ) - ^ + - ••+ — + • • • .

with Z, = Z,(/7).

§2.1. Complex Manifolds

37

Thus ^ is a pole of f{p) a n d / ( g ) = 00. Considering/:/?->/(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{<^~^{ Wx)n Uj) c W^ into the domain W^ = u^{W^)^U''.\^ for any pair j , A with ~n W;^) ^ ^ T^ 0 , O;,, is C\^ is called a C^ map where 1 ^ r ^ oc. If m = n, O maps the domain Dc=2 homeomorphically onto a domain E cz T, and both O and 4>~^ 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)
§2.2. Compact Complex Manifolds

41

1, 2 , . . . , m, Z2 can be used as a local coordinate on C C intersects with the line ^o = 0 at the m points q/c = (0, e"''", e^-^^/m) ^^^j^ fc = 1 , . . . , m. We use w = Z^o/^i = 1/^1 as a local coordinate on C in a neighbourhood of q^. Consider the Abelian differential o) = dzi on C Since l + zl" + z^ = 0, we have z^""^ (izj + z^"^ dz2 = 0. Hence, cu = -z^'^z]'"" dz2 in a neighbourhood of Pfc. Thus p/c is a zero of order (m - 1 ) of w. In the neighbourhood of q^, 0) = -w~^ dw, hence q^ is a pole of order 2 of co. Hence the divisor of! of (o is given by m

m

f = (a,)= X ( m - l ) p , - I 2q,. fc=l fc=l

Thus 2 g - 2 = deg! = m ( m - l ) - 2 m , that is, g = | m ( m - 3 ) + l. Example 2.4. Let C be an algebraic curve in P^ defined by {CiCi- C0C3 = ^0^2 ~ f?= ^2-^1^3 = 0}. Let (^0, ^1) be the homogeneous coordinates of P^ Then the map (^0, ^1) -> (^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. <^s is m-to-one on ^^ + f r + ^T ^ 0, and one-to-one on ^J' + ^P + ^ r ^ O . Let C be the algebraic curve in P^ defined by ^o-^iT-^CT = 0, Then S is an m-fold branched covering of P^ with C as its branch locus of order ( m - 1 ) . Then denoting by xi^) the Euler number of a manifold M, we have x(S) = Substituting

A'(P^) = 3

mx(P')-{m-l)x{C).

and x(C) = 2-2g = m(3-m), x(S) = m(m^-4m

we obtain

+ 6).

In general, let Af" be a complex submanifold of a complex manifold W= W. Then for given qeM, we can choose local coordinates w^: p^

42

2. Complex Manifolds

^q(p) = (^lj(p)^ U(q) such that

" , ^q(p)) of W with centre ^ on a coordinate polydisk

MnUiq)

= {peU(q)\w-^\p)

= -" = w",{p) = 0}.

Then the map p -»(wl(p),..., w'^(p)) gives local coordinates of M centred at q. Let f{p) be a holomorphic function defined in a domain D of H^". Then the restriction fj^ip) of f(p) to M is a holomorphic function in M n D. For, if we represent/(/7) by a holomorphic function of ( w ^ , . . . , w^) in a neighbourhood of ^ G M n D as /(p)=/.(w^,...,0, we have /M(/^)

= / , « . . . , <,0,...,0).

Let / ( p ) be a meromorphic function on W". For any qe W, we can choose a sufficiently small L/^(g) such thsit f(p) = hq{p)/gq(p) on C/(g) where hg(p) and g^C/?) are relatively prime holomorphic functions. If U(qi)n 1/(^2) ^ 0 , by (2.13) there is a non-vanishing holomorphic function u(p) on U(qi)n U(q2) such that gq,{p) = u{p)gqj^p) there. If ^ G M , the restrictions hq^ip), gq^iip) ofhq{p),gq{p), respectively, to M are holomorphic in M n U(q). If gqMip) vanishes identically for some qe M, then for all q^M, gqMip) vanishes identically. In fact, for qi, q2e M with Mn U(qi)nU(q2)9^0, we have gq,M(p) = UM(p)gq,M(p) on MnU(qi))n ^(^2)) where UM{P) is non-vanishing. Therefore if gqjM(/?) = 0 identically, by the analytic continuation we obtain gq2M{p) = ^ identically. If gqM(p) does not vanish identically, we have /M(/?) =

hqM(p)/gqMip)'

Thus given a meromorphic function fi p) = hqip)/ gqip) on W, its restriction fMip) to a submanifold M c: W'^ is a meromorphic function on M unless gqip) vanishes identically on M. Suppose given an algebraic manifold M = M"" c: p". Let P(f) and QiC) be homogeneous polynomials of the same degree. We call/(^) = PiO/ QiC) a rational function. Since a rational function / ( ^ ) is meromorphic on P", the restriction fMiO to M is a meromorphic function on M unless Qi() = 0. In particular iCklCJ)M is a meromorphic function on M unless Cj = 0 on M. Let M be an algebraic manifold, and qeM. We may choose meromorphic functions z^ip),..., z'^ip) on M such that (Zg(/7),..., 2^(7?)) give local coordinates with centre q. Proof We may assume that M n ^ 0 ^ 0 , and that qe Mn %LQ. Let iz\,..., z^) = iz\ip),..., z^'qip)) be the non-homogeneous coordinates of

§2.2. Compact Complex Manifolds

43

p G ^0- We choose local coordinates w^: p ^ [w^,...,

w^) = w,(p) such that

MnU{q) = {p\w'^^' = --- = w^ = 0}. Since 3(Z',...,2-,...,Z")

a(w;,...,w-,...,w,")^"' after renumbering Zj if necessary, we may assume that

det-h"

^ ^ 0

at^.

Put z^ = z^(p)-z^(q) for /c= 1 , . . . , m. Then since ( w ^ , . . . , w^) are local z^(/?)) are also local coorcoordinates on M with centre g, /? ^ (z^ip),..., dinates on M with centre q, and each Zq(p) is extended to a meromorphic on M. I function UJCO)M-z^iq) Thus there exist abundant meromorphic functions on an algebraic manifold.

(b) Quotient Space Let \y be a complex manifold. By an automorphism of W, we mean a biholomorphic map of W onto itself. In other words, an automorphism is a map which does not alter the complex structure of W. When we define the product of two automorphisms gi, g2 of W by their composite gig2, the set of all automorphisms of W forms a group, which we denote by ^. The unit of ^ is the identity of W^ and the inverse of g e ^ is the inverse map g~^ of g. Any subgroup of ^ is called a group of automorphisms of W. Let G be a group of automorphisms of W. For peW, the set Gp = {g(p) \geG} is called the orbit of G through p. Two orbits Gp and Gq do not have a common element unless they coincide. Thus W is decomposed into the mutually disjoint orbits of G. The set of all orbits of G is called the quotient space of W by G, which we denote by W/ G We may consider that W/ G is obtained from W by identifying pe W with qeW if there is an element g G G such that q = g(p). Example 2.6. Let \ ^ = C"^^ - ( 0 , . . . , 0), and (^o, • • •, ^ J coordinates on C"^\ We denote by C* the multipHcative group of all non-zero complex numbers. Any gGC* defines an automorphism of W via

44

2. Complex Manifolds

Thus C* becomes a group of automorphisms of W. An orbit C*(^o, • • •, ^n) of C* on W is a complex line f on C"^^ with the origin deleted. Hence In the sequel we explain a method of constructing a compact complex manifold as a quotient space of a given complex manifold. In general, the quotient space W/ G of a complex manifold is not a complex manifold. In order that W/G may be a complex manifold, G must satisfy certain conditions. Let g be an automorphism of W. A point p of W is called a fixed point of g if g{p)= p. We say that G is fixed point free if any ge G except the identity has no fixed point. Definition 2.7. An group G of automorphisms of W is called properly discontinuous if for any compact sets Ki, K2 of W, there are only a finite number of elements ge G such that g ( / c i ) n i ^ 2 ^ 0 If G is properly discontinuous, each orbit Gp is a discrete subset of W. Theorem 2.2. Let G be a group of automorphisms of a complex manifold W. If G is fixed point free and properly discontinuous, W/ G has a canonical structure of a complex manifold induced from that of W. Proof Put W = WIG, and p = Gp. For each qe W, choose local coordinates Zq'. p -^ Zq{p) = {zlj{p),..., Zq{p)) wlth ccntrc q. Take r > 0 such that the closed polydisk { ( z ^ , . . . , z^) | |z^| ^ r, /c = 1 , . . . , n} is contained in the range of Zg, and put Uriq) =

{p\\zl,{p)\
Given qe W, if we choose r sufficiently small, we have g{Ur{q))n Ur{q) = 0 for any ge G except the identity. In fact, if otherwise, there is an element g„ e G, g„ 7^ 1, for each n = 1 , 2 , . . . , such that g„( t/„) n l!7„ # 0 , where 11^ = Ur/niq)' Then gn(Ui)n Ui9^0 for any n. Since [ t/i] is compact, and G is properly discontinuous, { g i , . . . , g„,...} must be a finite subset of G Therefore we can find at least one g„ say, gi such that gi (L^„) n l/„ 7^ 0 for infinitely many n. Then gi{q) = q, hence, ^ is a fixed point of gi 7^ 1, which contradicts the assumption. Consequently for sufficiently large n, g(Un)r) U„ = 0 for any ge G with g 7^ 1. Writing simply r instead of r/n, we have g{Ur(q)) n Uriq) = 0 as desired. Thus for sufficiently small r > 0, g( UXq)) n U^q) = 0 for g G G, g # 1. Hence Gpr\ UXq) = {p} for any pe UXq). Consequently the map p^p is injective on Uriq). By the definition of the quotient topology, a subset U of W is open if and only if its inverse image by the above map p^p is open in W. Thus Ur{q) = {p\pe UXq)} is an open set, and the map p^p maps Uriq) homeomorphically onto Uriq). Thus W is di topological manifold.

§2.2. Compact Complex Manifolds

45

For each point q^W, fix a coordinate polydisk Uriq) with r = r(q) satisfying the above condition. Then by Theorem 2.1, we can choose coordinate polydisks Uj=Uj(qj) with 0
^OT

peUj

is a homeomorphism of Uj into C". Suppose O/O L4 7^0, and take an arbitrary point PQE Ujn U^ with PQ e Uj. Then there is a unique element ge G such that gpo€ t4, and in a sufficiently small neighbourhood of po, we have ^j(^) = ^qjiP).

Zkip) = ZqkigP),

P G Uj,

Since g is biholomorphic, the map Zj(p)^ Zj^(p) is also biholomorphic. Thus {zi, Z2,...} forms a system of local complex coordinates on W, and defines a complex structure on \y. I Thus if G is properly discontinuous and fixed point free, the quotient space W= W/G becomes a complex manifold. The map p^p, which is represented as ^qj(P)^^j(P)

= ^qj(P)

on Uj

in terms of the local coordinates defined above, is a holomorphic map of W onto W, and biholomorphic on each coordinate neighbourhood Uj. We express this fact by saying that 1^ is a covering manifold of W. We call G the covering transformation group of W over W. If there is a compact subset K of W such that W= K = {p\pe K}, W is compact. Thus, taking a suitable group of automorphisms G of a complex manifold W, which is properly discontinuous and fixed point free, we can construct a new compact complex manifold W/ G. Note that K= W if and onlyif \y = U , . o ^ ( ^ ) Example 2.7. Put W = C and fix a complex number (o with Im w > 0. Let G be the group consisting of all parallel translations of the form g^„: z -^ z + mo) + n, with m,neZ. G is properly discontinuous and fixed point free on W. Let F be the parallelogram in C with vertices 0, 1, a> + 1, and co. Then LJ gmn(F) = ^' Consequently C = C/G is a compact Riemann surface. C is obtained from F by identifying x, 0 ^ x ^ 1, with x-\-(o, and ^o;, 0 ^ ^ ^ 1, with t(o-{-\. Hence C is diifeomorphic to a torus, and the genus of C is equal to 1.

46

2. Complex Manifolds ii

0)

(O + l

X + O)

« tcoJ

/tco + l

F

0

^

1

X

Figure 1

As is well known in the classical theory of elliptic functions, C is represented as a cubic curve in P^. We explain this briefly below (for details, see, for instance, [12], Absch. II, Kap. I). Put (o^„ = m(o-^n, and define

mn/

P{z) is a meromorphic function on C, called the Weierstrass P-function. P(z) is G-invariant, that is, P(z + w^„) = P(z) for any comn- Hence P(z) is considered as a meromorphic function on C = C/G. P{z) has a pole of order 2 at each w^„, and holomorphic elsewhere. Therefore as a meromorphic function on C, P(z) has a pole of order 2 at the point 0 e C corresponding to OG C, and holomorphic elsewhere. P(z) satisfies the following differential equation: r ( z ) ^ - 4 P ( z ) ^ + g # ( z ) + g3 = 0,

(2.15)

where the coefficients g2 and g3 are given by

g2 = 60

I

^-,

g3=140

I

^ .

In view of (2.15), put P{0 = ^0^?-4^2 + g2^0^2 + g3^0.

Since gl-27gl 9^0, it is easy to see that for any ^ e P ^ , at least one of the partial derivatives of P(f) does not vanish. Hence P(f) = 0 defines an

§2.2. Compact Complex Manifolds

47

algebraic curve T in P^. Consider the mapping O of C into P^ defined by 0 : z ^ ^ = (p(z) = ( l , P ' ( z ) , P ( z ) ) . is clearly holomorphic except at 6. Since P(z) has a pole of order 2 and P'{z) has a pole of order 3 at z = 0, writing (z) as (z) = (z^ z^^'(z), z^^(z)), we see that <|) is also holomorphic at 6. It is obvious from (2.15) that ^{C)czT.By more detailed investigation of the properties of P(z), we can prove that O is a biholomorphic map of C onto T. Thus C may be identified with the cubic curve T in P^ as a Riemann surface. We call C an elliptic curve. If we put a = e^'"'^, then e^^'('"^+") = ct'". Therefore via the holomorphic map z^w = e^^'\ which maps C onto C*, grnn induccs an automorphism g* of C*: w-^a'^w. Put G* = { g * | m 6 Z } , which corresponds to G via the above map. Since l m w > 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


+00

b^^zUi

and

./.(z)= I

h,k = 0

c.zUi

h,k = 0

be their power series expansions. Then

l{a';a2r'"K,z';z^

/ ( z ) = lim

Let n be the minimum of |al'a2l for all pairs (h, k) with ^,fc= 0, 1,2,..., such that not both b^k and Ch^ are not zero. Then

Z/

/ ( z ) = lim

h

k/

\~-mi

-^^+^L{0CX0L2/M

h k

ChkZiZ2

Since lim^^+oo («I'<^2/M)~'" == 0 for laj^il > 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^<27r, and A = 1 , . . . , i^, be all the distinct numbers among these a^a2/fx, and put PAz)=

I

hkzUi

and

QAz)=

Z

c^.z^z^'.

Then

/ ( z ) = lim ^

.

(2.17)

A= l

Since/(z) is meromorphic, some QA(^) are not identically zero. Suppose, say, Qi(z) is not identically zero. Then from (2.17) we obtain

Qi(^)

We say that a^ and a2 are independent if a\^ a2 for any pair (/z, k) of natural numbers.

52

2. Complex Manifolds

(i) The case that a^ and a2 are independent. In this case since there is a unique pair (/i, k) with a\a2 = /JLCI, we have Pi(z) = bhj^z\z2 and Qi(z) = Chk^\^2 foJ* this (/i, k). Hence / ( z ) = hhkl^hk is a constant. Thus there is no non-constant meromorphic function on M = W/ G. (ii) The case that aj and a2 are not independent. Let {p,q) be the pair of natural numbers such that a\ = a^ and that p is minimal among such pairs. Then \i a\ = 0:2, {K k) = (mp, mq) for some natural number m. Therefore letting (/105 ^0) be the pair such that a\^a2^ = /mei and that ho is minimal among those pairs satisfying the same condition, we see that any such (h, k) is written as /i = /10+ ^A ^ = ko-mq for some non-negative integer m. Then

m

where we denote h^ for fe^^ and c^ for c^^. Since a:f = a | , z f / z | is Ginvariant, hence, gives a meromorphic function on M = WIG^ a n d / ( z ) is a rational function of z\lz\. Thus a meromorphic function on a Hopf manifold M = WIG of dimension 2 is either a constant if a^ and ^2 are independent, or a rational function of z\l Z2 if «i and 0:2 are not independent. From this result we see again that M is not algebraic. For n ^ 3, too, if a ^ , . . . , a„ are independent, that is, afi • • • ajj" ^ 1 for any n-tuple (/ci,..., /c„) ^ ( 0 , . . . , 0) of natural numbers, there are no non-constant meromorphic functions on M = WIG. (c) Surgery We explain another method to construct complex manifolds. Let M be a complex manifold, and 5 c M a compact submanifold of M. We construct a new complex manifold M = {M — S)u S by replacing S by another compact complex manifold S as follows: Take domains W and W^ such that 5 c: Wi c [ \ ^ J c: \y c: M, where [ W] is assumed to be compact. Let 5 be a compact complex submanifold of a complex manifold W such that there are a domain W] with 5 c: W, c [ W'j] c: W, and a bihojomorphic map O of W - 5 onto W - 5 such that ^{W^-S) = W^-S. Let M be the manifold obtained by glueing M-[VV^i] and W by identifying pe W-[Wy] with p = <^~\p)e W-[Wi] via : M = ( M - [ T ^ i ] ) u W. Since 0 is biholomorphic, M becomes a complex manifold. 77IM5 M f^ a complex manifold obtained from M by replacing W by W: M =

(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|<£, (Z, [W]) = (Z, [Z,]) = (Z, [ZZ2]) = (Z, [ZW]), hence,

which implies that ^ j i V^-> M-Sc: M and ^2^ V2^ ^ ^ ^ coincide on V^n V2. Therefore defining ^ = '^i^on V^, and ^ [ ^ ^ ^ on V2, we have a biholomorphic map ^ of N onto M. Thus M = N is a Hopf manifold. Example 2.12. Let M = C",^5 = 0 = ( 0 , . . . ,^0) e C", and 5 = P"~^ We define a complex structure on M = (M — S)u S as a complex submanifold of Qn^pn-i -j^ ^1^^ following manner. Since each point ^ of P"~^ is, by

56

2. Complex Manifolds

definition, a line ^ in C" through 0, ^ x f G C " x P " ~ ^ is a hne in C " x ^ through the point 0 x ^. Put M = U^X^-{(z,f)GC"XP''-^|zGn. Let ( z i , . . . , z„) be the standard coordinates on C", and ( ^ i , . . . , ^n) the homogeneous coordinates on P " " \ In terms of these coordinates, we denote a point (z, ^) by (zi, .^.., z„, ^ i , . . . , ^„). Then z G ^ if CjZj, - ZJCR = 0 for7, k = ! , . . . , « . Therefore M is an analytic subset of C" xP"~^ defined by ^,Zfc-4z,- = 0,

j\k=l,..,,n,

(2.18)

Since P"~^ is covered by n pieces of coordinate neighbourhoods Uj = {fEP"~'|{,-7^0},7 = l , . . . , n , C " x P " - ^ is covered by C"xU^s. To verify that M is a submanifold C " x P " ~ \ consider Mi = M n C " x L/^i. Let ( w 2 , . . . , Wfc,..., w„) be the inhomogeneous coordinates on Ui where w^ = 4 / ^ , . Then on C" x U^ (2.18) is equivalent to ^k = WfcZi,

/c = 2,3, . . , n .

(2.19)

Thus Ml is a submanifold of C" x L^j. Similarly each Mj = MnC xUj is a submanifold of C" x Uj. Therefore M is a complex submanifold of C" x

Let O be the restriction to M of the projection (z,^)-^ z of C" xP"~^ onto C". Then O is a holomorphic map of M onto M = C", and O"^(0) = OxP""^ Consequently, to see that M = ( M ^ - 5 ) u S where 5 = "^(0) = OxP"~^ and 5 = 0, it suffices to show that 5 is a submanifold of M and that ^ maps M — S biholomorphically onto M — S. By (2.19), we have Ml = {(Zi, W2Zi, . . . , W„Zi, 1, W2, . . . , Wj I (Zi, W2, . . . , W„) G C"}. Therefore (zj, W2,..., w„) gives a local coordinate system of M defined on the coordinate neighbourhood Mi. We define Mi similarly. Then M is covered by these M i , . . . , M„. In terms of these local coordinates, 5 n Mi is a submanifold of Mi defined by the equation Zj = 0. Similar results hold for Sn M 2 , . . . , Sn M„. Thus 5 is a submanifold of M. Let (z, ^)j= (zj, ...,z„,^)eM-S. Since z G ^, and z ?^ 0, ^ = ( z i , . . . , z„). Hence on M — 5, is given by O: ( z i , . . . , z„, Z i , . . . , z„) -> ( 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<5o. It is easily verified that QqiP^) is biholomorphic to Mi given in Example 2.10.

52.3. Complex Analytic Family (a) Complex Analytic Family In the definition of an elliptic curve C given in Example 2.7, there appears an arbitrary parameter co on which the complex structure of C depends. Similarly the complex structure of a Hopf manifold given in Example 2.9 depends on the parameters a j , . . . , a„, while in the definition of P" no arbitrary parameter appears. Thus the complex structure of a complex manifold M often varies as the parameters t = (t^,..., t^) which appear in its definition varies. This being the case, we say that the complex structure of M depends on t. We write M, for M if we want to express the dependence on t explicitly. Let us consider how a compact complex manifold M^ depends on parameters t. First consider a complex-valued function f(t)=f(ti,...,tn) defined in a domain B^C^. We may consider/(O as a complex number varying as t moves in B. If f(t) is holomorphic, f{t) is said to depend holomorphically on t In this case the graph o f / ( O ^=Uf(t)xt

=

{ifit),t)eCxB}

t&B

is a submanifold of C x B, hence, a complex manifold. Similarly we may consider a complex manifold M^ depending on ^ as a "function" of teB, but unlike a function f{t) above, there is no space containing all compact complex manifolds. Nevertheless, we can consider the set M={J

MfXt

teB

corresponding to ^ = U f : 3 s / ( 0 x ^ above. Thus we reach the following definition. Definition 2.8. Suppose given a domain B in C", and a set {Mt\teB} of complex manifolds M^ depending on t = (ti,..., t^) e B. We say that M^ depends holomorphically on t and that {Mt\te B} is a complex analytic family of compact complex manifolds if there is a complex manifold M and a holomorphic map m of M onto B satisfying the following conditions.

60

2. Complex Manifolds

(i)

m~^(t) is a compact complex submanifold of M.

(ii) Mt = (iii)

m-\t).

The rank of the Jacobian of m is equal to m at every point of Ji.

Here in (iii) by the rank of the Jacobian of m, we mean the rank of the Jacobian matrix in terms of local coordinates. Thus, let (z\,..., z^, z^^\ . . . , Zq^"^) be local coordinates on M and let (^1,. • •, ^m) = ^iz]j,...

,Zq,z-;.^

z

Then (iii) impHes that

^^

^^(7^

n

n+ 1

n+ m x - ' ^ -

Therefore by §2.1(d) we can choose a system of local complex coordinates { z j , . . . , Zj,...}, Zj'. p-^ Zj{p), and coordinate polydisks % with respect to Zj, satisfying the following conditions. (i) (ii)

Zj(p) = (z](/7),..., zj{p), ^ 1 , . . . , tj, (tu-^>,tj n = {%\j = l,2,...} is locally finite.

= m(p);

Then

{p-^(z'j(p),...,z^{p))\%nM,^0} gives a system of local complex coordinates on M^. In terms of these coordinates, m is the projection given by m: (zl...,

z;, ^ 1 , . . . , t^)-> (tu . . . , tm).

For 7, A: with Ujn Uj^9^ 0 , we denote the coordinate transformation from Zfc to z, by fjk'. ( z L . . . , z;;, 0 ^ ( ^ j , . . . , ^;, 0 =y;fc(^L..., ^L t). Note that ^ i , . . . , ^^ as part of local coordinates on Ji do not change under these coordinate transformations. Thus fjk is given by ^r =fjk{zl,...,

Zfc, ^ 1 , . . . , t^),

a = l,...,n.

(2.21)

In what follows by a complex analytic family, we mean a complex analytic family of compact complex manifolds unless otherwise mentioned. We sometimes denote a complex analytic family {Mt | ^ G B} by (Jt, B, xu), where Ji, B and m are as above. We call t its parameters, and B its parameter space. Definition 2.8 is obviously extended to the case B is an arbitrary complex manifold.

§2.3. Complex Analytic Family

61

Example 2.13. The set {C^\lm(o>0} of elliptic curves C^ = £/ G^ forms a complex analytic family where G^ = {m(o + n\m, neZ}. In fact, put B = {(oeC\lma>>0}. Then a group of automorphisms of C x B defined as G = {gmn' (z, a))-^{z + ma)-\-n)\m,

neZ}

acts in a properly discontinuous manner without fixed point. Therefore M=CxB/G is a complex manifold. Since the projection (z,(o)^a) of CxB to B commutes with g^„, it induces a holomorphic map xu of M onto B. We have

m-\(o) = CX(o/G = C/G^ = C^. Using (z.o)) as local coordinates on M, we easily see that the rank of the Jacobian matrix of m is equal to 1. Thus {C^\(oe B} forms a complex analytic family. Definition 2.9. Let M and N be two compact complex manifolds. N is called a deformation of M if M and N belong to the same complex analytic family, that is, if there is a complex analytic family (Ji, B, m) with a complex manifold B as its parameter space such that M = ttT~^(?o) and N = vi7~\ti) for some ^o, U e B. Two complex analytic families {Ji, B, m) and (^, B, TT) are called holomorphically equivalent if there is a biholomorphic map <^ of Jt onto Jf such that V7 = 7r°^. This being the case, O maps M, = V7~^{t) biholomorphically onto Nt = rr~^(t), hence M^ and Nt are biholomorphic. As in the case of complex manifolds, we often identify two holomorphically equivalent complex analytic families. Let M be a compact complex manifold, and B an arbitrary complex manifold. Then {M x B, B, vi) forms a complex analytic family where m denotes the projection to the second factor. If a complex analytic family {Ji, B, nr) is holomorphically equivalent to (M xB, B, m) above with M = m'^ito) for some ^o^ A {^, ^,'^) is called trivial. If {M, B,m) is trivial, Mt = tiT~\t) is biholomorphic to M for all te B. In this case the complex structure of M^ is independent of t. A trivial family may be considered as an analogue of a constant function. Let (Ji, B, trr) be a complex analytic family, and U SL subdomain of B. Let Mu = m~\U), and ury the restriction of m to U. Then {My, U, vju) forms a complex analytic family, which we call the restriction of {M, B, m) to U. If {Mu, U, vTu) is trivial, we say that {M, B, m) is trivial over U. Theorem 2.3. Let {Ji, B,xu) be a complex analytic family of compact complex manifolds, and to any point of B. Then Mt = TxT~^{t) is diffeomorphic to M^ = m'^ito) for any t e B.

62

2. Complex Manifolds

Thus the differentiahle structure of complex manifolds does not change under deformation. To prove this theorem, we need to prepare some results from the theory of differentiahle manifolds. First let M be an arbitrary differentiahle manifolds, { x i , . . . , Xj,...} a system of local coordinates on M, and % the domain ,xf) = Xj{p). We assume that {%|7== (1,2,...} is a locally of A^: p^{x],... finite open covering of M. For7, k with %r\%Lk^0, let Xj =fjk{Xj,

. . . , Xfc )

be the coordinate transformation. Consider a smooth curve 7: t-^Xj{t) = ( x j ( 0 , . . . , < ( 0 ) on Ji. We denote {d/dt)xf(t) by xf{t). Then t;, = (i;j,...,i)7) = ( x j ( 0 , . . . , x f ( 0 ) is the tangent vector of y at Xj{t).

Under the coordinate transformation x]" =ffk{Xk), the tangent vector of 7 is transformed as follows.

vf=l'-^vt=lfivt

(2.22)

Let <^(x,) be a continuously differentiahle function on M. Then we have d

"^

fi

^^(^^•('^)=.?,<^^(^^('))Consequently the vector Vj corresponds to the differential operator

a=1

dXj

63

§2.3. Complex Analytic Family

Since by (2.22) (2.23) a=\

the operator X^ ^f(d/dxf) ates. If a tangent vector

a=l

oXj

aXfc

does not depend on the choice of local coordin-J

m

v(xj)= S t;;(x,) — a =1

OXj

is assigned to each point Xj of J^, we call v(Xj) a vector field. If all i^/(x,), a = 1 , . . . , m, are C°°, the vector field i;(x,) is called a C°° vector field. Suppose given a C ^ vector field v{xj) such that v(Xj) T^ 0 at every point of Ji. Then r/ie system differential equations, -xf

= vf(x],...,

O ,

a = 1, 2 , . . . , m,

(2.24)

has a unique solution xJ = xj{t) under any given initial conditions ^;(0) = ^r,

a = l,2,...,m.

We denote this solution by xj{t, ^,). Then x^{t, ^,) is a C°° function of t, ^ ! , . . . , ^r. Since the system of equations (2.24) are invariant under coordinate transformations by (2.23), i75 solution xf(t,^i) gives a smooth curve r-» x,(r, ^,) = (xj(r, ^ , ) , . . . , x7(r, ^,)) on M starting from the point ^, of M.

Figure 7

Next, let M bQ Si difierentiable manifold, B SL domain of R"", and tzr a C°° map of M onto B. We assume that (Ji, B, t^r) satisfies the following conditions.

64

2. Complex Manifolds

(i) (ii)

Mt = xJiT~\t) is compact for any t e B. the rank of the Jacobian of m is equal to m at every point of M.

Then, as in the case of a complex analytic family, we can choose a system of local coordinates { x i , . . . , x^,...}, Xji p^Xj(p), satisfying the following conditions. (i) Xj(p) = (x]{p),..., xjip), h,..., tj, ih, ...,tj = m{p). (ii) {% |j = 1, 2,...} forms a locally finite open covering of Jt where % is the domain of Xj. Assume that 0 e J5, and take an open cube U = {t\\ti\
onto V7~^{U) such

tZT o ' ^ = TT.

Proof. We use induction on the dimension m of B. (1°) The case m = l. In this case B and U are open intervals in R: U = (-r, r), and [-r, r ] c R We denote ( x j , . . . , xJ, t^) simply by (xy, ^i). First let us construct a C°° vector field on J^ which is given on each % as

i "/(^>'')i+^a^l

oXj

(2-25)

all

We denote the vector field d/dt^ on ^ ^ by (d/dti)^. Then with respect to the coordinate transformation ^7 —fjki^k, h) —f^ki^k, • • •, ^fc, ^i), we have

Let {pj(Xj, ti)} be a partition of unity subordinate to {%}. Then lPjixj,t,)(j^y

(2.27)

is a C°° vector field on M. On ^^, we have

fc

\oti/k

ct = \ k9^j

otx

dXj

\dti/j

§2.3. Complex Analytic Family

65

M

B

_d_

Figure 8

where we write p^ for p^Cx^, t,). Therefore the vector field given by (2.27) has the required property (2.27) if we put

k^j

dti

Consider the system of differential equations for this vector field: ^ - ^ j dti

It

i^j, "',Xj,

t^),

a = 1 , . . . , n,

(2.28) = 1.

For any point ( 4 0) = Ul...,

^r, 0) G Mo = m-\0),

K = ^;(40,

let

a = l,...,n,

Mt) = t be the solution of (2.28) satisfying the initial condition xnO) = ^r, ^i(0) = 0.

a = l,...,n,

66

2. Complex Manifolds

Then t^(x]Ui,t),...,xJi^,,t),t),

-r
is a smooth curve through the point {^i, 0) on MQ. Denote the point (^i, 0) simply by ^, = (^J,..., ^D, and the point ( ( 4 0), 0 ) on Mo x t/ by ( 4 0Then

gives a diffeomorphism of MQ X (7 onto m~^{ U). Clearly m o '^((^j, t)) = t =

^((4 0). (2°) General case. By identifying mutually diffeomorphic differentiable manifolds as usual, it suffices to prove that m~^{U) is MQX U and that m coincides with the projection TT, which we express simply by writing vT-\U) = MoXU. Put

w

:{(ri,...,r^_i)|ki|
and

U^ =

{tm\-r
Then we have U = U"^'^ x 11^. Since by induction hypothesis m~\ U""'^) = MQ X t/'""^ in order to prove that m~^( U) = MQ X U, it suffices to show that For VT(P) = {ti,..., tm-u tm) with/7G M, define tzr^ by m^{p) = t^. Then ti7^ is a C°° map of M into [R which maps vj~^{U) onto (7^. We denote by TTm the projection of m~^{U"^~^)xUm onto L4t. As in (T), using a partition of unity, we can construct a C°^ vector field on M of the form Z

<(xj,...,x;,r„...,r^)-\+-^.

By solving the corresponding system of differential equations dx j-= v^'ix],. . . , x;, ^ 1 , . . . , tj, ~dt

a = 1 , . . . , M,

du = 0, dt

^=l,...,m-l,

I dt

-=1,

we obtain a diffeomorphism ^ ^ of m~\U'^~^)xU^ onto tu~^(i7) such that xjjm°'i^m = ^m' Thus we have m~\U) = X!T~\U'^~^) x U^. Hence we obtain m-\U) as desired.

I

= m-'iV^-')

xUm = MoX U"^'' xUm = MoXU

§2.3. Complex Analytic Family

67

Applying Theorem 2.4 to a complex analytic family (Jt, B, m) WQ obtain the following theorem. Theorem 2.5. Let (M, B,m) he a complex analytic family^ c an arbitrary point ofB, and Mc = VT~\C). For a sufficiently small coordinatepolydisk U{c) with centre c, there is a diffeomorphism ^ ^ of Mc x U{c) onto m~\{U{c)) such that m°'^c — "n-c where TTC is the projection of Mc xU(c) to U{c). I The diffeomorphism ^ c maps each M^ x t onto M, = rjj~\t) for t e U(c). Thus for te U(c), M, is diffeomorphic to M^. Given any two points ^o, t of B, we can choose a series of coordinate poiydisks L^(co),..., U{ci) as in Theorem 2.5 such that toe (7(co), te U{ci) and that l[7(c,_i)n U{cj)^0 for J = 1 , . . . , /. Therefore M^ = tJT~^(0 is diffeomorphic to M^ = m~^{tQ), which proves Theorem 2.3. I Now consider holomorphic vector fields on a complex manifold. Let M be a complex manifold, { z i , . . . , z^,...}, zy. p -> {z],..., z]) = Zj{p), a system of local complex coordinates, and Uj the domain of z,. A vector field on M is called a holomorphic vector field if it is represented on each coordinate neighbourhood Uj as

a=l

OZj

with holomorphic functions vf(p) on U,. On L^n 1/^7^0, we have X t;;(p)—= I a=\

VUP)-^,

a=\

dZj

OZk

that is, " az

f;(p)= ^ I= 1 <5Zfc ^^ffcp). Given two holomorphic vector fields v{p) = lvfip)^ a

and OZj

wip) =

lwf{p)^, a

dZj

we define their linear combination by "

cMp)-^C2w(p)=

d

X (civ]'ip) + C2w]'{p))---^ a=l

with Ci, C2GC.

OZj

Then the set of all holomorphic vector fields on M forms a vector space.

68

2. Complex Manifolds

(b) Examples Let {Mt \teB}bQ 3. complex analytic family. The structure of M^ may vary continuously or discontinuously as t varies continuously. We give some examples in this subsection. Example 2.14 First consider the complex analytic family {C^lcoeW^} of elliptic curves where IHl^ = {w GC|lm w > 0 } . Let C^ = C/G^. There are infinitely many choices of generators of G^. Two elements (Oi = aa)-{-b, and generate G^ if and only if ad-bc = ±l: 0)2 = ceo-{-d of G^ with a,b,c,deZ G^ = {mcoi + no)2\m, n eZ}. Interchanging Wi and C02 if necessary, we may assume that ad - be = I. By the coordinate transformation z-^ z'= z/co2, coi and C02 are transformed into co' = co 1/0)2 and 1 = 0^2/^2 respectively. Clearly Im a;'>0. Conversely, if ao) + b co)-\- d'

a,b,c,deZ

with ad-bc=

1,

(2.29)

Q = C / G ^ and C^' = C/G^' are biholomorphic to each other.

Figure 9

Conversely, if C^ and C^' are complex analytically equivalent, a biholomorphic map of C^ onto C^^ is lifted to a linear map z^z' = az-\-l3 of C with a 7^0. Then ao) and a must generate G^. Thus o)'= aao)-\-ba, with a J - fee = 1. Therefore a>' is and 1 = cao) + da for some a,b,c,deZ given by (2.29). Thus Q is biholomorphic to C^^ if and only if (2.29) holds. The set of all linear transformations of the form (2.29) forms a group of automorphisms of H^, which we denote by T. T acts in a properly discontinuous manner on H"^. F = { a > | - | ^ 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 "^ * * * ~

<^20-^l

<^21^1

^22"

Thus fi(zi) must be a quadric az^+bz,+ c in Zj:

dzi

(2.35)

dzi

Since a holomorphic function on P Ms a constant, a holomorphic vector field on Mo = P^ xP^ is given by (fl,z?+^zi + ci)::^+(ai^? + i8i^i + 7i) ^ 5Zi

a^i

Consequently there are six linearly independent holomorphic vector fields on MoNext we consider holomorphic vector fields on M ^ = L / i X P ^ u U2XP\m^l. From (2.35) a holomorphic vector field on Ui xP^ has the form (2.36)

i^i(^i)r-+(«i(^i)^?+i8i(zi)^i + ri(^i))7?-, d^i az,

where Vi(zi), ai(zi), Pi(zi), 7i(zi) are entire functions of z^. Similarly a holomorphic vector field on U2XP^ is given by (2.37)

^^2(^2)—-+(«2( ^2)^2 + i 8 2 ( Z 2 ) ^ 2 + 7 2 ( ^ 2 ) ) — , dZ2 d^2

where 1^2(^2), ^2(^2)* l^ii^i), 72(^2) are entire functions of Z2. For a holomorphic vector field on M^ which has the form (2.36) on Ux x P ^ and the form (2.37) on U2y^P\ we must have Vx{z,)^+{a,{z,)C\

+ P,{z,)Cx +

y,{z,))^

dZi

dCi

= V2(z2)^'^(a2(z2)^l 0Z2

+ P2(z2)^2^y2(z2))-:^ 3^2

(2.38)

on (7i X P^ n (72 X P^- Since Zj = I/Z2 and ^1 = z^^2 by (2.32), we have ( d

I d

m-l.

— =-——+mz2

az2 IH2

Z2 dzi ^^ dC,

d

af 1 zTdCx

2 d

d

^2—=-^1—+mzi^i—,

dzi

dCi

75

§2.3. Complex Analytic Family

Substituting these into the right-hand side of (2.38), and comparing the coefficients, we have

ri(2i) = -;;rr2

(t)-

From these equalities, we have fi(zi) = azf+bzi + c, ai(zi) = Zfc=o ^k^u Pii^i) = -mazi + d, and 7i(zi) = 0. Therefore on M^ with m ^ 1, there are (m + 5) Hnearly independent vector fields corresponding to the arbitrary constants a, b, c, CQ, CJ, . . . ,c^ and ^ above. Thus the number of linearly independent holomorphic vector fields on Mrn is 6 for m = 0, and m + 5 for m ^ 1. Hence if m = n (mod 2), and mT^ n, Mni and M„ are not biholomorphic.

Chapter 3

Differential Forms, Vector Bundles, Sheaves

As was stated in the Preface, the purpose of this book is to explain the development of the theory of deformations of complex structures since 1956. In this chapter we prepare various results about complex manifolds required for this purpose.

§3.1. Differential Forms (a) Differential Forms on Differentiable Manifolds Let 2 be a differential manifold, { x i , . . . , x,,.,.}, Xji p ^ Xj{p) = (x],..., xj") a system of local C°° coordinates on 2, and Uj the domain of Xj. We may assume that the open covering U = {U,|j = 1,2,...} of S is locally finite as in Theorem 2.1. Also as in §2.1(a), we may consider S as obtained by glueing the domains % = Xj{Uj)c:U'^^ by identifying pe Uj with its local coordinates Xj = Xj{p): 2 = U j %. Thus in the sequel we write Uj for %•. Then we have S = U^ ^ with Uj c R'". Differential forms on S, their wedge products and exterior differentials are defined as follows. Let cpj^,..., (pj^ be real or complex-valued functions on Uj. Suppose given on each Uj a linear combination of dXj's

X (pja dxf = (pji Jxj + • • • + (pj^ dxf. a= \

If on Uj n f/fc, the equality m

I

m

f>j.dx^= Z ^^dxl

(3.1)

(p^iPj.dx'j + '-^iPj^dx'P

(3.2)

holds, we call

§3.1. Differential Forms

77

a differential form of degree 1 or simply a 1-form on E. If cpja, a = 1 , . . . , m, are continuous, continuously differentiable, or C°°, cp is called continuous, continuously differentiable, or C°°, respectively. Equation (3.1) implies that by substituting

J8 = l C/Xfc

we have

Also we mean by (3.2) that

where we put

/3 = 1 C X

Similarly we call 1

m

1 m


(Papdx'' Adx^=-

X (pj^p dx]" A dxf

a 2-form on 2, where t/x" A dx^ = -dx^ A C/X". More generally, we call (^= —X (pcc.-adx""' A • • ' Adx"'an r-form on 2. If the coefficients ^a,..a, are continuous, continuously differentiable, or C°°,

Z

<^a,-«. dx"' A • • • A ^X"^

ai<---
In the following, we use this convention unless otherwise mentioned. Writing a, p,..., y for a j , a2, • • •, <^r, we have 9 = — X <^a-.-'y ^^" A • • • A dX^. r!

78

3. Differential Forms, Vector Bundles, Sheaves

We define a linear combination of r-forms cp^ and ijj as

r! The wedge product of an r-forni (p and an 5-form il/ is defined as follows: Cp A ll/=

X <Pa,--a^B,---B

^ ^ " ' A ' * ' A dx"''

A dx^'

A ' ' ' A dx^\

(3.4)

For a continuously differentiable function / on S, its differential

dx^

dx""

is a continuous 1-form. For two such functions / and g, clearly

d(fg)=fdg-hgdf holds. For a continuously differentiable r-form
(Pa--ydx'" A' ' ' Adx'^,

we define its exterior differential dip by d(p=—

X d(p^...^Adx" A' ' ' Adx^.

Thus, for a 1-form (p we have

a ft dX^

2 cx,i3 \dX

dx'^/

and if

In general for an r-form (p, if we write d(p as ^^ = ,

. , , . I ( ^ < P ) a o a , - a . ^ ^ " ° A ^ X " ^ A - ' ' A C?X^

(r+1)!

(3.5)

§3.1. Differential Forms

79

we have

(rf
(3.6)

oX ^

s=0

The definition of exterior differential d is independent of the choice of local coordinate systems. Namely, on Uj n U^, we have di — Y,(Pja..ydxf A- ' ' A dxj] = dl — Y. (Pka-y dxt A ' •

'Adxlj

First we prove this for r = 1. Since by (3.3) ^dxi A

oXj

we have d(pjc. = Z
But since d^x^/dxf dxf is symmetric, and dxf A dx" is skew-symmetric in a and /3, we have A dxf = I -

I d(^) a

\dXj

J

p^a OXj

^ rfxf A dxf = 0,

(3.7)

dXj

hence .dx^k

X d(pj^ A dxf =1^ dcpkx AY.—^ a

A

A oXj

dxJ =1^ d
Proof for general r also follows immediately from (3.7). The wedge product cp AIJJ is clearly bilinear in cp and (/^. If cp is an r-form, we denote it by (^'^ explicitly. Then the following equality holds:

ip'Ar={-'^rr^^'^

o.s)

Clearly d is linear, that is d{cx(p + €2^1^) = Ci d(p + C2 di//,

where Ci, C2 are constant. Taking the exterior differentials of both sides of (3.4), we obtain d((p' A (A') = d(p' A (A' + (-1) V A dil/\

(3.9)

3. Differential Forms, Vector Bundles, Sheaves

80

since dil/p^.^p^Adx""' A ' • • A(i3c"'- = (-l)''^x"i A- • ' Adx'"'- Adif/p^...^^. If (p is C^, then d^(Pa,:.aJdx°' dx^ is symmetric in a and (3, hence by the definition of the exterior differential (3.5), we have ddcp= dl-Tl ^-^^^^dx"" a dx \r\

A t/x"> 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)

<5(X/c, . . . , Xfc )

In case S is orientable, by choosing a system of local C°° coordinates {xi,X2,...} such that (3.11) holds on each L^n{7fc#0, we define an orientation on 2. Let {wi, M2,...} be another system of local C°° coordinates satisfying (3.11) on each W^ n W^ 7^ 0 where W^, is the domain of w^. Then for all pairs (A,7) of indices such that W^ n t/^ 7^ 0 , det(aMA/^^f) are all simultaneously >0, or <0. In the former case we say that {u^} defines the same orientation as {x^}, and in the latter case the inverse orientation. Let 2 be an orientable differentiable manifold oriented by a system of local C°° coordinates {x,,X2,...}. Let a; be a continuous m-form with m = dim 2. Then, on each coordinate neighbourhood Uj, w is written as (o=—y ml

(Oj

dxf' A • • • Adxf^. •'

'

Since in this case { a i , . . . , a^} is a permutation of { 1 , . . . , m}, we can write (o as (O

= (Ox...mdXj A • • • A

dxf.

On UjH Uk7>^0, we have (Ojx...^dx]

A • • • A dxj" = 0)kx-mdx\A

• • • A dxj^.

§3.1. Differential Forms

81

Using ^xf = Xr=i (d^k/dxf)

dxf, we see that

dxi/\'-'AdXk=

Z

T;:^***73r^^/A---AJx-

djXk, . . . ,Xic)

" ^^^^77^

,1

T^^^J

, m

A • • • A dxj ,

d V A j , . . . , Aj ;

thus Jxjt A • • • A cixIT = d e t ^ ^ 4 ' 4 ^ ^ - ^ ^ ^ ^ ^ ] A • • • A ^ x f .

(3.12)

d{Xj, Xj, . . .^ Xj )

Therefore we have d{Xk, . . . , X ^ ) CO^l...^ = ^ ^ t " : ; r i 7;^(^kl-rrf d(Xj, . . . , Xy j

(3.13)

The integral of co on 2 is defined as follows. Let {py|j = l , . . . } be a partition of unity subordinate to a locally finite open covering U = {Uj\j =1,...}. Then we define

I

Pjco,...^ dx] dxj ' • • dx^.

co=l

(3.14)

If E is not compact, we must examine the convergence of the sum in the right-hand side of (3.14). Since by hypothesis det(axfc/axf ) > 0 , it follows from (3.12) and (3.13) we have \(Oji...m\ dXjA- • -Adxf = |a>fci...^| dxlA'

' • AdXk

on Ujn Uk9^ 0 . Hence, by putting \aj\ = |(Wyi...^| dXj A • • • A dxj^

on each LI, we obtain a continuous m-form \a)\ on S, and 1^1=1 Js

Pikji

ml dx] • • • JxJ".

j J Uj

The sum in the right-hand side of this equality either converges or diverges to +00. If J^ |fo|<+oc, the right-hand side of (3.14) converges absolutely, hence J^. (o exists. In this case we say that the integral J^. co converges absolutely. When Js |w| = +oo, we don't define J^ co. If the orientation of the differentiable manifold 1 is fixed, the integral Jj- 0) defined above is independent of the choice of systems of local C°°

82

3. Differential Forms, Vector Bundles, Sheaves

coordinates {xj, X2,...} and partitions of unity {pj\j=l,...}. In the definition of the integral we introduce a partition of unity, in order to, we must represent the integral of w as a sum of continuous functions defined on domains of 2. If we use the theory of Lebesgue integral, we can do without a partition of unity. Namely, if we choose Lebesgue measurable subsets Vj cz UjJ = 1,2,... such that V; n V^ = 0 for jy^k and that U^- Vj = 1, then w=X J2

j

(Oi...mdXj • • • dxj". J Vj

We say that an r-form cp = (p{p)=—

^

(pj^ ^dxf A'

"AdxJ

' "5---»7

vanishes at /? G S if all coefficients (Pja---y vanish at p. We define the support of
L

d(p = 0.

(3.15)

Proof. Choose a system of local C°° coordinates {Xj} such that each Uj is an interval {x, e W" \ \x]\
[ d
Js

\j=l

= i \ d{pj,). 7 = 1 J L/j

I

Therefore it suffices to prove \^ d{pj(p) = 0 for each Uj. We write m Pj

where o-a = o-a{Xj) is supp o-^ c= Uj. Since

dx'jA'"A

a

dxf-'

continuously

d(Pj
a=ldXj

A dxf^'

A • • • A dxj",

differentiahle

^dxJA'-'AdxJ',

function

with

§3.1. Differential Forms

83

we have r

"^ r f)(T

d{pj^)=l

-f^dx]---dxT.

Then it is easy to see that each term of the right-hand side of this equahty is 0. For example, the first term vanishes because

i

dxj---dxT[a^il,x%...,xJ') -1

-a,(-\,xj,...,xj')]

= 0.

I

Corollary. Let (p be a C^ r-form on 1 and ij/ a C^ (m-rthat at least one of supp (p and supp if/ is compact, then d(pMjj = {-\y-'

I)-form. Suppose

(pAdii^.

(3.16)

Proof. By (3.9), d{(p AII/) = d(p A (/^ + ( - 1 ) V A dif/. Then applying (3.15), we obtain (3.16). I Any open subset \y of E is itself a differentiable manifold. Hence the above argument applies also to W. We call (p a closed differential form if <^ is C^ and dcp = 0. An r-form (p on a domain W is called an exact differential form if there is a C^ (r — 1)-form ijj on W such that (p = dif/. If both cp and i/^ are C \ and cp = dil/, then dcp = 0. This is obvious from (3.10) if if/ is C^. In case r/^ = C \ this is proved as follows. Proof Since the problem is local, it suffices to see d(p — ddif/ = 0 on each Uj, For this it suffices to see that J^r ddij/ A 77 = 0 for any C°° (m - r - l)-form rj on Uj with compact support. Since by (3.10) ddr] = 0, using (3.16) we obtain ddij/A T] = {-ly J Uj

\ dil/Ad7] = - \ il/Addrj = 0. J Uj J Uj

I

Thus C ^ exact differential forms are closed. Locally its inverse also holds. Theorem 3.2 (Poincare's Lemma). Suppose that a C ^ r-form (p, r^l, satisfies d

84

3. Differential Forms, Vector Bundles, Sheaves

Proof. We fix r and prove the theorem by induction on m for m^r. For simpHcity we only treat the case r = 3 below, but the generalization is straightforward. (1°) The case m = r = 3. Put (p = (p2i2dx^ Adx^ Adx^, and define il/ = i/^i2 dx^ A dx^ b y ^12=4^12{X\X^,X^)

=

'P312(X\X^,X^)

dx\

0

Then if/ is SL C^ 2-form on U and dil/=—- dx^ A dx^ A dx^ = (P3i2ix\ x^, x^) dx^ A dx^ A dx^ = (p. (2°) The case m>r = 3. In what follows the indices a, jS, y represent numbers from 1 to m - 1. Define ^ = ^^1

^a^dx-Adx^

by


Then J/^ is a C^ 2-form. By (3.6) we have

dx"^

ax"

'

-^

dx^

—^=\Cl(p)mccf3y

dx

= ^,

hence mafi

ax"

dx"

dx"

& dx- = ^„,,(x) -
Jo

ax'

and Kdllj)rnaP=-—=(p^acfi{x).

Therefore putting 1


m-l

I

3 ! a,/3,y=l

9ap^(x',...,x'"-',0)dx"Adx^Adx^

§3.1. Differential Forms

85

we obtain dif/=

(p — q>".

Since (p and cp" are C\ dij/ is also C\ hence, by the above result, d(p" = dip - ddijj = 0. Since (p" is a C ^ 3-form of m - 1 variables x \ . . . , jc'"~\ by the induction hypothesis there is a C^ 2-form 1

m-l

r=-, I K,{x\...,x'"-')dx-Adx^ such that dif/"=(p\ Therefore cp = d{il/ + il/"). If (p is C°°, if/ and <^" are also C°°. Then since by induction hypothesis, we can choose ij/" to be C°°,
..,X"')G

17, then for any e^: 0 ^ ^« ^ 1, a = 1 , . . . , m, we have

(b) Differential Forms on Complex Manifolds Let M be a complex manifold, { z , , . . . , z,,...}, z^: p^ ^jip) == ( ^ j , . . . , z") a system of local complex coordinates, and Uj the domain of z^. As stated in §2.1(a), by identifying peUj with its local coordinates Zj = Zj{p), we may regard M = U^. %, where % = Zj{ Uj) is a domain in C". In view of this fact, we write % also as Uj in the sequel. By Theorem 2.1, we may choose Uj such that U = {Uj\j = l,2,...}is locally finite, and that each Uj is apolydisk defined as Uj = {zj 6 C" ||2J|
|z;| < r;}.

(3.17)

instead of z", we may assume that each Uj is a polydisk of ^• = { z , - G C " | | z j | < l , . . . , | z ; | < l } .

(3.18)

Let 2 be the underlying differentiable manifold of M (see §2.1(f)). For the system of local complex coordinates {zy}, Zj = {z],..., z"), by decomposing z/ into the real and imaginary parts as zj" = x]""'^ + /x^^", a = 1 , . . . , n, we obtain a system of local C°° coordinates {xj, X2,...}, Xj = ( x j , . . . , x^"). Then by (1.41) we have

( 9 ( X f c , . . . , Xfc")

d{zl...,zl)

>0.

(3.19)

86

3. Differential Forms, Vector Bundles, Sheaves

Consequently 2 is orientable. We define the orientation of 1 by the above {Xj}, and call it the orientation of the complex manifold M. Consider a differential form (p on M: (p = —X (Pj^.^...^.^ dxj' A • • • A d x / ,

(3.20)

where the coefficient (Pjvyv, is a complex-valued function. Since

xf-i=.i(z; + z;) and xf =^^(z;-z;), we have dxf-'

= k ^ ^ ; + dz"^) and

d x f = — (dz/ - dzj).

Substituting these into (3.20), we obtain
-^-.l
p+ q =

dzf'^'''^dzf\

rplql

We assume that (Pja^---oc ^^•••^ are skew-symmetric in the indices a i , . . . , a^ and ^ 1 , . . . , j8q separately. We call a differential form

p\q\

I <^,a,-«,^-^-#, ^^7' A • • • A ^Z/P A Jzf • A • • • A dzf^

a differential form of type (p, q), or simply a (/?, q)'form. Since the coordinate transformation: Zfc->Zy is biholomorphic, we have

Hence ^/le property that cp is a (/?, q)-form does not depend on the choice of local complex coordinates. If (p is a (/?, ^)-form, we often write it as (p^^'^^ for the sake of distinction. An r-form (p"^ is uniquely expressed as

Let / be a C ^ function in a domain of M. Then

a = l dZj

a = l OZj

§3.1. Differential Forms

87

We define 5 / = 1 -^dzf

and 3f= I

a = l OZj

-^dzf.

a = l OZj

Then we have df=df+df. We can verify by easy calculation that df and df do not depend on the choice of complex coordinates. For a C^ (p, ^)-form _1_

^iP,ci)^

y <^,«,...«,^-,...^-, Jz;> A • • • A dz-^ A dzf^ A • • • A dzf^,

we define ^9^^'"^ = - 7 - ^ 1 a^,«,...«,^-....^, A Jz;> 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

(^<^^^''^0,•«o«^••«.^-r••.-.= I: (-1)

Similar formulae hold for d(p^^''^\ For an arbitrary C^ form (p =Zp,<j
a^=:^a^(^'^)

and 5<^ = X ^^p^'''''^.

Then clearly we have d=d-^d.

(3.21)

oo

3. Differential Forms, Vector Bundles, Sheaves

Also the following formulae are immediately obtained from the definitions of d and d. d{(p' Ailf) = d ' • A dip,

(3.22) (3.23)

Suppose that d(p^^'^^ and dcp^^'"^^ are both C\ Then we have

Considering the type of each term in this equality, we obtain 5(99^'''^^ = 0,

aa(;p^^'^^ + aa(p^^'^> = 0

and

ddcp^^'"'^ = 0,

hence the following equalities hold: dd = 0,

dd = 0

and

dd = -dd.

(3.24)

A (p, 0)-form (p = {l/pl) ^ (Pja,---otp dzf' A • • • A dzfp is called a holomorphic p-form if the coefficients
A J z / i A • • • A t/z/p = 0

is equivalent to ^
^ZJ A J z j A • • • A dz] A ^ z j .

(3.25)

Since any domain W of M is itself a complex manifold, the above results hold also for differential forms on W. In what follows we chiefly treat C^ differential forms. A C^ differential form (p is called a ^-closedform \{'d(p — 0. Let (p = (^^^'^^ be a C°° (j?, ^)-form on M or on a domain of M with ^ ^ 1. If there is a ^ ° ° ( A
§3.1. Differential Forms

89

In order to prove this theorem, for cp and if/ we put <^a,...a, = — I
'A dz\

^ - » - p ^ ( ^ _ l ) ; ^ ^«,...«,^-,..-^-, dz^^ A'-'A

and

dz\

Then (pa^-'-ap is a (0, g)-form, and il/a,---a^ is a (0, g - l ) - f o r m . Moreover ai(^ = 0 is equivalent to ^^a,...a^^O, and (p=dil/ is equivalent to <Paj...«p = '^^ayccp' Hence it suffices to show Theorem 3.3 for a (0, ^)-form. First we show Lemma 3.1. Let f{z) be a C^ function on \z\
^dCAdl

(3.26)

Then ^_g(z)=f{z), dz

\z\
(3.27)

holds. Proofi Take an arbitrary a such that 0 < a - < p . We shall prove (3.27) for |z|
fAQdCAdc ,

2771 J Ji^i^p

. . ^ = 1,2,

(-Z

we see that g(z) = gi(z) + g2(z). Since f2iO = 0 in |^| ^ cr, g2(z) is a holomorphic in \z\ < a, hence (9g2(z)/az = 0.

90

3. Differential Forms, Vector Bundles, Sheaves

Defining/i(f) = 0 in U\^R, with w = re'^. Then we have

we extend/i(^) on all of C. Let ^ = z + w,

i f f

dwAdw

ITTI J JC

=- -

W

\\fi{z+re'')e-'Urde.

Consequently gi(z) is a C°° function of z, and we have

)Z az

TT J J

_ L f r a/i(z+ w) 277/ J J c ITTI J J c

^^

dZ

(iw A dw

^^

>^

dz w) t/w A Jw (9/i(z+

dw

W

27r/ J J,f

•i^llK^.<-»>f)-

Since |z| p + cr. Therefore by Green's theorem, the last integral is equal to - lim

= lim—r

Mz +

dw

w)—=Mz),

>+o27nJ.rwi

vv

where W^^ = {w||w|< e}. Thus for |z|
S

I

and let


q^l,

/3i<---
be a 5-closed C°° (0, ^)-form on UR which does not involve the differentials dz'"'^\ . . . , dz". Given an arbitrary real number p with 0 < p < i?, we prove

§3.1. Differential Forms

91

by induction on m the existence of a C°° (0, q -~ l)-form

j8,<---
such that (p=dil/ on Up. Since for p^m-tl,

we have

— cpp,..^^(z\ . . . , z " , . . . , z") = (d
(3.28)

(Pp^...p^{z\ . . . , z"^, z'"^\ . . . , z") is holomorphic in z'"^\ • . . , z", hence a does not affect on z'^^\ . . . , z". The proof given below is an analogy of that of Poincare's lemma. As before we consider only for the case q = 3, but the generalization is straightforward. Let

where cp^py are holomorphic in z"'^\ . . . , z". Take an arbitrary cr with p
Z

^a/3 ^ Z " A dz^,

by

1

'ma/3VZ , . . . , Z

,fe, Z

, . . . , Z »

—^^

2-77/

^

-

d^ A a ^

By Lemma 3.1, if/ is C°° in U^-, and

dZ

and by (3.28), we have

ar

= 0,

for ^== m + 1 , . . . , n,

hence in case m = q = 3, dil/ = a(iAi2 dz^ A (iz^) = —zr dz^ ^ dz^ A dz^ = (p^n dz^ A dz^ A JZ^ = cp. dZ

92

3. Differential Forms, Vector Bundles, Sheaves

In case m> q = 3, we have

+ x<(3
\ dZ

dZ

dZ

/

Consequently putting (p"= (p-dip, we obtain a ^-closed C°° (0, 3)-form (P"=

Z


By induction hypothesis, there is a C°° (0, 2)-form

r=

I rai.dz'^Adz^ a/3 '

a
on Up such that (p" = dil/" on L/p. Hence on L^p, we have

which accomplishes the induction. Thus, putting m = n, WQ see that for an arbitrary p with 0 0 , there exists a C°° (0, ^ -2)-form ;^2 on L'p^-e such that 5;^2 = ^3 ~ 0"2- Let A(r) be a C°° function of r with r > 0 such that A(r) = 1 for r ^ p i , and that A(r) = 0 for r ^ p 2 - 2 e , and define a C°° (0, g-2)-form X2 = Xii^) on Up^ by -(^)^/Ar2(^)A(|z|),

\z\
lo, Then we have fA'2(z),

^ ^ ^ ^ ^ =lo, -

|z|^P.-2..

93

§3.1. Differential Forms

Pi

Pi

P3

PA

P5

Figure 1

Put 4^3 — ^3 — 8x2' Then ip^ is a C°° (0, g —l)-form on Up^ such that d4f3 = dil/3 = (p, and that 1^3 = ^As -dXi = 'As -dXi = 4^2 on Up^. Similarly we can construct successively C°° (0, g-l)-forms (/^^ on Up^, /c = 2, 3 , . . . , such that aj/Jfc = <;P on Up^, and that (A/c+i = '^k on ^ , ^ _ j . Let ifj be the C°° (0, q - l)-form on UR obtained by putting ip = if/]^ on each Up^, for /c == 2, 3, Then dil/ =

fk(z)-

Sz'r

(z-)-".

Since this series converges absolutely and uniformly on Up^_^, we can choose a sufficiently large / such that

Pkiz)=

Z

a „ , _ „ ( z ' ) ' " ' • • • (z-)-"

mi,...,m„=0

satisfies |A(z)-P;,(z)|<^,

|z|
(3.29)

94

3. Differential Forms, Vector Bundles, Sheaves

If we put

4(z) = Mz) - P^iz)

P,_,{z),

/c = 3 , 4 , . . . ,

then ifki^) is a C°° function on t/p^, and, since (Afc+i(z)-(Afe(z)=A(z)-Pfc(z)

on |z|
(3.30)

we have on |z|
|0^/c+i(z)-(Afc(^)l<^

Hence the sequence {lAk(^) | ^ = m, m + 1,...} converges absolutely and uniformly on each ^,^_,, m = 2, 3,4, Put (A(z) = lim i}if,(z). k^oo

Then, by (3.30), we have 00

^(z) = ,^„(z)+ I

(A(z)-Ffc(z))

fc=m

on L/p^_i, where iAm(^) is C°°. Moreover since each/fc(z) - Pk(z) is holomorphic, and Zr=m (/fc(^)~^fc(^)) converges absolutely and uniformly by (3.29), this sum is holomorphic on Up^_^. Consequently ip = il/{z) is C°^ and dilj = dil/m = (p on Up _^. Therefore if/ is C°° and dil/ = (p on (7R = U°° (7 I

§3.2. Vector Bundles (a) Tangent Space Let M be a C°° manifold, { x j , . . . , x^,...}, Xj'. p-^ Xj{p) = ( x j , . . . , xj") a system of local C°° coordinates on it, and Uj the domain of Xj. Moreover we assume that Vi = {Uj} is locally finite. As stated in §3.1(a), we identify Xj = Xj{p) with p, and ^j = Xj{Uj) with Uy, respectively. As stated before (p. 62), for a smooth curve on M passing through p, its tangent vector at /7 as a linear differential operator is written as m ^

a=1

a

^

dXj

§3.2. Vector Bundles

95

We call such v a tangent vector of M at p. Let C^iM) be the IR-vector space of all C°° functions on M. Then as a linear differential operator v defines an IR-linear map of C°°(M) to IR as follows: v:^^v{ilj)= I vj-^{p), a -1

For il/,(pe C^iM),

il^eC^iM).

(3.31)

oXj

we have

v{ilf(p) = ik(p)v(
^(Xj)= Z

xf^^iXj),

where ^a(xj) is a C°° function of x^. In fact, putting tXj = (tx],..., we obtain ^(x.)=

-MtXj)dt= Jo

at

Z x; a=l

txj"),

-^(rx,)^^. J o C'^j

Hence we may put

*-<''>=rii which is clearly a C°° function of x,. Take p € C°°{M) such that /o(p) = 1, and that s u p p p c [/(/>). Since il>{p) = 0, i;(pV) = f(
P V = X px;p^„(x,)

96

3. Differential Forms, Vector Bundles, Sheaves

on Uj. We denote the natural extension of pxf and p'^^iXj) to all of M by the same notation px" and p^^iXj) respectively. Then by (3.32), we have m

K
putting vf = v(pxf) we obtain (3.31).

I

In view of this result, we may adopt the following "intrinsic" definition of a tangent vector. Definition 3.1. A linear map v of C°^{M) to R satisfying (3.32) is called a tangent vector of M at p. This is the modern definition of a tangent vector, which does not involve local coordinates, but it is not so easy to understand at first glance. For a tangent vector

a=1

OXj

at pe Uj^ M,v],..., vf are called the components of v with respect to the local coordinates Xj. Suppose pe Ujn ^4, and let f L • • • ? ^IT be the components of v with respect to x^. Then the following equations hold: _,

^

dXj

o

^

dXj

dXj ,

,

,

,

This is the transformation rule between the components of a tangent vector with respect to Xj and x^. The R-vector space consisting of all tangent vectors of M at p is called the tangent space of M at /?, and is denoted by Tp{M). Tp{M) is an m-dimensional IR-vector space, and the map v-^{vj,..., vj") maps Tp(M) isomorphically onto U"". The union of all Tp{M), peM, is called the tangent bundle of M and is denoted by r ( M ) : r ( M ) = UpeM Tp(M). Since a tangent vector at/? 6 Uj is written as u, = ( i ; j , . . . , i;j") in terms of local coordinates, T(Uj) = UpeUj Tp(M) is the set of all pairs (/?, Vj) with /? G UJ and tJy G R"": T(Uj) = {(p,Vj)\peUj,VjeR-}=UjXU-, Thus r((7y) /za5 ^/le structure of the direct product Uj xlR"". Hence we have T(M) = UT(Uj) = UUjXU'",

(3.34)

§3.2. Vector Bundles

97

where for pe Ujn U^, by (3.33) (/?, Vj) and (/?, v^) are the same point of T{M) if and only if ^j-

^ Aiv)vt -idxi

a = l,...,m.

(3.35)

Thus T(M) may be regarded as a differentiable manifold obtained from glueing up Uj xW^ via the identification (3.35). Each Uj xR'" is a coordinate neighbourhood of T(M) and ( x ] , . . . , xj", vj,..., vJ") gives the local C°° coordinate system on UjXW^. Since the projections UjXW^^ Uj and UkXR""^ U^ coincide on Uj x R'" n Uk xR"", they define the projection 77:r(M)->M. In terms of local coordinates, TT is written as ^•.{x],...,xT,v],...,vT)^{x],...,xT), hence

TT

is C°°. Moreover clearly

TT~^{P)=

Tp{M).

(b) Vector Bundle Definition 3.2. Let M be a differentiable manifold. A differentiable manifold F is called a vector bundle over M if it satisfies the following conditions: (i) (ii) (iii)

A C°° map TT: F-^ M of F onto M is given. For every peM, 7r~\p) is a z^-dimensional R-vector space: 77-~^(^)sR'', where P is independent of p. For every qe M, there exists a neighbourhood U, qe Uc: M, such that TT" ((7) = (7 xR^, and that for any peU,pxU'' is isomorphic to 7r~\p) as an R-vector space: TT~'\p)=p xR^

We call 7r~^{p) the fibre of F over p, and denote it by Fp. v is called the rank of F. M is called the base space of F, and TT the projection. We also denote F by (F, M, TT) if we want to make explicit its base space and projection. Let U = {L^|j = 1,2,...} be a locally finite open covering of M, and (F, M, TT) a vector bundle over M. If each Uj is taken sufficiently small, then by (iii) above, we have 7T-\Uj)=UjXU'^

= {{p^^j)\peUj,^j

=

{ej....Aj)^^l-

Since pxW — TT~^{P), ( ^ j , . . . , ^/) is a system of coordinates on the vector space 7r~^{p). ¥ov pe Uj n U^, ( ^ j , . . . , ^J) and {^\, -- - Ak) are two systems

98

3. Differential Forms, Vector Bundles, Sheaves

of coordinates on the same vector space 7T~\p), hence there is an invertible linear transformation (/j^^) such that ^j=i

f?Jc,(p)d,

a = 1,2,...,

p.

(3.36)

p= i

Namely (p, ^j)e Uj xW and (p, ^J^)G U^ XW are the same point on F if and only if (3.36) holds. We call (f j , . . . , ^J) the fibre coordinates of (p, ^j) e Fp over Uj. Since the subdomain 7r~\ Uj) of M is, by (iii) above, isomorphic to Uj xR" as a differentiable manifold, these f j , . . . , ^J are C ^ functions on 7T~\Uj). Consequently in (3.36), ^f is a C°° function of (p, fL • • •, ^k), hencQfjlp(p) is a C°° function on Uj n U^. By using vector notation, (3.36) may be written simply as ^j=fjk(p)-^k,

(3.37)

where fjk(p) denotes the matrix {ffkp{p))<x,p=i,2,...,v' This is the transition relation between the fibre coordinates, fj^ =fjk{p) is called the transition function. Of course we have det fikip) # 0. A vector bundle F is determined uniquely by giving fjk,j, k—l,2, Hence we write F = {fju}Transition functions satisfy the following conditions. First we have fjjip) = 1, where 1 denotes the unit matrix. Next, since for any ^ / G R " , ^J = Mp)^i MP)

=fjk(p)fki{p\

= fjdP)^k

= fjkip)fkiiP)^h

peUjnUkn

U,

(3.38) we have (3.39)

Putting l=j in (3.39) we also obtain fjk(p)fkj(p) = l-

(3.40)

Conversely, given a locally finite covering Vi = {Uj} and C°° functions MP) = ifU(p))a,p = i,...,. on each UjnUk^0 satisfying (3.39) and (3.40), we can construct a vector bundle F = [Jj UjXW over M by identifying ( A 4 ) e C4XR" with (pjjk{p)^k)e UjXW" for pe UjnUk9^0. Let F and G be vector bundles over M. If there is a diffeomorphism h of F onto G such that h maps each fibre Fp linearly onto Gp, F and G are called equivalent. In this case, take a locally finite open covering {L^} of M with sufficiently small Uj, and let ^^ = ( ^ j , . . . , ^J) and 17^ = (77J,..., 77J) be the fibre coordinates of F and G over U/ respectively. Then h is written on 7r~\Uj) as ^: ( A I/)-^ ( A ^,) = (P, hj(p)ij),

§3.2. Vector Bundles

99

where each component of the matrix hj{p) = (hfpip))^^^^^ ,, is a C°° function of p G UJ, and det hj(p) ^ 0. Let ^fc(/?) and gj],{p) be the transition functions of F and G respectively. Then as is easily seen, we have &fc(p) =

hj{p)fjk{p)h(py

peUjn

(3.41)

Uk.

Thus we may consider that G is nothing but F with another fibre coordinates Vj - (vh • • •' Vj) obtained from ^j = ( ^ j , . . . , ^J) by applying the transformation (/?, ^j)^{p, rjj) = (p, hj{p)^j). In this sense we often identify F and G if they are equivalent. M x R'' is a vector bundle over M. If F is equivalent to M xR'', F is called trivial Let F be a vector bundle over M, and TT its projection. A continuous map a: p-^ o-{p) of a domain Wc: M into F is called a section of F over W if a{p)e Fp for all pe W. In terms of fibre coordinates, o-ip) = (p, (Tj(p)), cTjip) = (cr'jip),...,

(rjip)),

peWn

Uj,

o-(M)

M

Figure 2

where o-jip) are vector-valued continuous functions satisfying crj(p)=fjk{p)'CTj,(p)

(3.42)

on WnUjn U^. If o- is continuously differentiable, C°°, etc., that is, each function (Tj{p) is continuously differentiable, C°°, etc., the section a is called continuously differentiable, C°°, etc., respectively.

100

3. Differential Forms, Vector Bundles, Sheaves

Comparing (3.33) and (3.42), we clearly see that the tangent bundle T(M) of M is a vector bundle over M with the transition functions fjk{p) = (fU(p)) where

A continuous vector field on M m

^

a=1

oXj

is nothing but a section of T(M). Here each d/dxf is a C°° vector field on Uj, hence a C°° section of T{M) over U^. For an arbitrary vector bundle F, using the fibre coordinates over Uj, put ej^:p^ej^{p)

=

(pAO,...,0,r,0,...,0))eUjXU\

Then e^„ is a C°° section over Uj, and any section cr of F over a domain \y is written as (r:p^crip)=

X o-f{p)ej^(p)

for

peWnUj.

We write this simply as cr= i

cr-(p)ej^.

(3.43)

By (3.36), we have ek(3= I f]kp{p)eja'

(3.44)

a =1

Let cTi,..., cr^ be C°° sections of F over a domain L/. {a-i,..., cr^} is called a basis of F over U if for each p^ U, {cri(/?),..., o-^ip)} is a basis of Fp = 7r~\p). For example, the above-mentioned {e^i,..., e,v} is a basis of F over Uj, and {(9/5xj,..., d/dxf) is a basis of T{M) over L^. Given a basis {cTi,..., o-^} of F over (7, every point of TT~^{U) is written uniquely as i

rcrAp),

peU,

(f^...,neR^

a =l

By writing this point as {p, f), with f = (^\ . . . , ^''), we obtain an identification of iT~\U) with UxU\ (i\ . . . , f'^) is called the fibre coordinates of

§3.2. Vector Bundles

101

( A f) with respect to the basis {o-i,..., a^}. In this notation we have ^«(/') = ( A ( 0 , . . . , 0 , r , 0 , . . . , 0 ) ) . Replacing IR by C and R'' by C in Definition 3.2, we obtain the definition of a complex vector bundle. In contrast to this, we call a vector bundle in the sense of Definition 3.2, a real vector bundle. For a complex vector bundle, its fibre coordinates Cj = (^j, • • •» ^/) ^ C'' form a complex vector, and the components of the transition function ffkfiip) are complex-valued C°° functions. Now consider vector bundles over a complex manifold. Let F be a complex vector bundle over a complex manifold M. If the transition functions fjkip)J, k=l,2,..., are all holomorphic, F is called a holomorphic vector bundle. Here by saying iha.t fjk(p) = {fjicf3(p)) is holomorphic, we mean that each component/^^^(p) is a holomorphic function of/? e L^ n U^. Then F is obtained by glueing up UjXC by identifying (/?, ^fc) G L4 XC*' with

(p,Q = {pJjdp)L)eUjXC'':F

= [JUjXC\ j

If fjk(p) is holomorphic, the map {p, Ck)-^{p, Cj) = {pJjk{p)Ck) is biholomorphic. Hence a holomorphic vector bundle F = {J. UjXC is a complex manifold, and the projection TT: F - > M is a holomorphic map. Let F be a holomorphic vector bundle. A section cr of F over a domain Wcz M is said to be holomorphic if it is holomorphic as a map of M into F. In terms of fibre coordinates, put o-\ p -^ o-j{p) = {(T]{P), . . . , crj(p)),pe Uj. Then o'J'(p) is a holomorphic function of p if a is holomorphic. In particular, ej.: p^ ej^(p) = (p,(0,...

,0,r,0,...

,0))

is a holomorphic section of F over U,, and (3.44) is also valid in this case. Let ( z j , . . . , z") be a local coordinate system on M, and Uj the domain of Zj. We assume that Uj is a domain of M and that Vi = {Uj} is a locally finite covering of M. Introduce local real coordinates Xj = ( x j , . . . , xj") by putting z" = x^"~^ + ixj"" as usual. Then {x„ . . . , x^,...} becomes a system of local C°° coordinates of the underlying differentiable manifold of M. The vector bundle of all tangent vectors of M which are written as

i vf^,

a=l

dZ

vfeC,

in terms of local complex coordinates, is called the tangent bundle of the complex manifold M and is denoted by T(M). The transition functions

102

(fU(p))

3. Differential Forms, Vector Bundles, Sheaves

of T(M) are given by fUP)=^ip)-

(3.45)

dZk

Hence T{M) is a holomorphic vector bundle. A holomorphic vector field on M "

d

«=i

oZj

is nothing but a holomorphic section of T(M). Note that T(M) is different from the tangent bundle r ( 2 ) of the underlying differentiable manifold S of M. We shall discuss the relation between them later. (c) Dual Bundle, Tensor Product In this paragraph, K denote either IR or C. Let E be a finite-dimensional vector space over K. As is well known, if {^i,..., e^} is a basis of E, any element ue E is uniquely represented as u= i

rea,

r^K

(3.46)

a=l

and the map u^(^\ ..., ^ ' ' ) G K"" gives an isomorphism of JB onto K"": E = K\ The set of all linear functions v = v(u) on E forms a vector space over K, which we call the dual space of E and denote by £*. If M e £ is written as in (3.46), we have

\a

/

a=l

Hence putting i7a = f (e„) and representing v by {rji,..., r}j,)e K"", we see that E'^ = K"" and that v{u)=Y.l^i CVcx- We write {v, u) instead of v{u). Then {v,u) = v{u)= i

rVa.

(3.47)

We call (v.u) the inner product of v and u. From (3.47) it is clear that E^^ = E. Definition 3.3. Let F be a real or complex vector bundle over a differentiable manifold M A vector bundle G over M is called the dual bundle of F if it satisfies the following conditions:

§3.2. Vector Bundles

(i) (ii)

103

For every point pe M, Gp is dual to Fp, i.e. the inner product (v, u) is defined between ve Gp and u e Fp. If we take a sufficiently fine locally finite open covering U = { m of M, then we can choose fibre coordinates ( f j , . . . , ^J) of F and (17J,..., 17/) of G over Uj such that for u = (p, ^ j , . . . , ^J) G Fp and f = (/?, 77J,..., 17J) G Gp, their inner product is given by {v,u)=

I ^^vja-

(3.48)

a=1

We denote the dual bundle of F by F*. Let fjk(p) = {fU{p)) be the transition functions, and f^ip) = {ffktip)) those of F*. Then we have

^;= z fU{p)ek.

vu,= 1 m(p)vja.

/3 = 1

a-1

Thus (v, u) = Xa ^JVjc = Z^ ^kVkp implies

ZZ/;".^(/^)ff^,«=ZZ/|^(p)^^^.«. Hence we have f%i{p)=f^jAp).

(3.49)

Therefore we obtain the following transition relation between fibre coordinates of F * : Vjcc= Z fkja(p)Vkp-

(3.50)

Note that we have F** = F. Now we review the definition of the tensor product of vector spaces. Let E and F be vector spaces over K, and let {^i,..., e^} and {/i,...,/.} bases of E and F respectively. Then the tensor product of E and F is the vector space with the set of all pairs (e«,/^) as its basis. We denote the tensor product of E and F by F ® F. Thus E®F consists of all elements of the form

w= z z r'ie^j,). r^GK For M = Z a ^"^« ^ ^j ^'^^ ^ "^Z^ V^fp ^ P^ we defitie u®v by

w®t;=zzr^^(ea,^).

104

3. Differential Forms, Vector Bundles, Sheaves

Then we have (^aj/g) = ^ a ® ^ , and any element w of E®F is represented as w=i

i r'e^®f,,

r'eK.

(3.51)

is bilinear. The map E xF-^ E(E)F given by (u,v)-^u®v Considering the dual space we verify that the tensor product E®F does not depend on the choice of bases used in the above definition. In fact for any te ( £ ® F ) * , (t, u®v) is a. bilinear function on £ x F . Conversely, any bilinear function b(u, v) on E xF is written as b(u, v) = (t, u®v) for some te (E®F)'^. Thus we naturally identify (E®F)* with the vector space of all bilinear functions on E xF. Consequently ( E ® F)*, hence also E®F = ( F ® F ) * * do not depend on the choice of bases of E and F. Definition 3.4. Let F and G be real or complex vector bundles over a differentiable manifold M. A vector bundle H over M is called the tensor product of F and G if the following two conditions are satisfied. (i) (ii)

For every p eM, Hp = Fp®Gp. Take a sufficiently fine locally finite open covering U = {Uj} of M and let ( ^ j , . . . , ^j^) and (rjj,..., rjj) be fibre coordinates of F and G over Uj respectively. Then for w = (/?, ^ j , . . . , ^j^) e F^ and v = (p, T/j,..., rjj)e Gp, the fibre coordinates of M®Z; are given by ^;^=^;^f,« = i,-..,M,/3=i,...,^: u®v = {p,^]vl-'-,^rvD'

(3.52)

We denote the tensor product of F and G by F ® G . Let the transition functions of F and G hQfjk(p) = {f]ky{p)) and gjkip) = {g%8{P)) respectively. Then since

^r-nf= I i fUP)g%B{p)avl the transition functions of F ® G are given by htAp)=f]Ly{p)g%s{p)^

(3.53)

The matrix hj^ip) = {hj^y^ip)) is called the Kronecker product of matrices fjkip)

and gjjc(p).

For a section a: p^o-{p) of F and a section r: p^rip) of G over a domain W^ M, p^o-(p)®T(p) is a section of F®G by (3.52), which is denoted by O-®T.

§3.2. Vector Bundles

105

If M is a complex manifold, and F and G are holomorphic vector bundles, F®G is also a holomorphic vector bundle by (3.53). If cr and r are holomorphic sections of F and G respectively, cr®T is a holomorphic section of F ® G . Let {Cji,..., gy^} and {fji,... ,fj^} be bases of F and G over Uj respectively. Then {Cj^ ®fj^ | a = 1 , . . . , /x, /3 = 1 , . . . , z^} forms a basis of F® G over Uj, and a section co of F®G over a domain Wc: M is represented as co=l

1 (ofip)

• ej^ ®fj,,

peWn

U,

a = \ (3 = 1

(d) Tensor Field Let (p = Z r = i ^jc^ ^^7^ ^jcc e C, be a 1-form on a differentiable manifold M at /?, and m

^

a=1

aX^-

a tangent vector of M at p. Introduce the inner product of cp and v by m a=l

Then the set of all 1-forms at p forms the dual vector space T^(M) of Tp{M) via this inner product. As in the case of T(M), UpeM '^pi^) has a natural structure of vector bundle over M. As is obviously seen from the transformation rule for 1-forms
(3.54)

with respect to the coordinate transformation x^, -> Xj, compared with (3.33), the vector bundle T'^iM) = [Jp^M TfiM) is the dual bundle of TiM), and the inner product {(p, v) given above is invariant under coordinate changes. A section of T'^iM) over a domain Wc: M is a continuous 1-form on W. {dx],..., dxf} forms a basis of T'^iM) over U, and the equality

holds. A 1-form

106

3. Differential Forms, Vector Bundles, Sheaves

The transformation rule (3.54) of this vector with respect to the coordinate change x^ -^ Xj is the same as that of the basis {d/dx]"} of T{M). In view of this fact we call {(pji,..., cpj^) SL covariant vector. On the other hand, the transformation rule (3.33) of the components of a tangent vector v = Za ^^i^/^^'j) is the inverse of that of the basis. Hence we call ( u j , . . . , i;J") a contravariant vector. Put (X) T''{M)=T''{M)®T''{M)®'

•

'®T''{M).

A C°° section o- of 0^^ T'^{M) is represented on each coordinate neighbourhood Uj as ^=

Z o-j^p...ydxJ®dxf®'

• •®dxj.

a,P,...,y

Here m'' coefficients cr^a/s y are C ^ functions on Uj, and for the coordinate change Xj -» x^ on U, n ^4, satisfy the following: _

^

dx^dxl

dxl

\,iji,...,v-^l oXj^ dX]^

dX]^

Given m'^ C°^ functions o}«...^ on each Uj satisfying (3.55) on UJD U^, we call o-ja...y a C°° covariant tensor field of rank r, and each a)«...^ its component with respect to the local coordinates Xj. For arbitrary local coordinates ( x \ . . . , x'"), the component of a covariant tensor field a^.-.y is given by

This is the classical definition of a covariant tensor field, which is, it is true, not intrinsic, but turns out to be very convenient. A C°° covariant tensor field of rank r is nothing but a C°° section of 0^^ T'^(M). r times

Similarly a section a of (x)' r ( M ) = r ( M ) ® T ( M ) ® • • '®T{M) resented on each Uj as

is rep-

and satisfies on [/, n t/t the relation dxt crr'= 1 •dl---Tf.<''\,...,v dXj

dXj

(3-56)

§3.2. Vector Bundles

107

We call orf'"^ a contravariant tensor field of rank r. With respect to arbitrary local coordinates ( x \ . . . , x""), o- is written as

Further, for example, a section cr of r ( M ) ® r * ( M ) ® r * ( M ) is written as

cr^^ is called a mixed tensor field, cr^^ satisfies the following relations:

A covariant tensor field o-^...y is called symmetric (or skew-symmetric) if it is symmetric (or skew-symmetric) in the indices a , . . . , y. Thus a covariant tensor field a^^ is symmetric if a^^ = a^^ and skew-symmetric if o'ap — ~(^/3a' We define symmetric and skew-symmetric contravariant tensor fields similarly. Let
A dx^

A'

' '

Adx^

be an r-form on M. Then (pai3---y is a skew-symmetric covariant tensor field. Since X (pa-.ydx"^®' ' '®dX^ a,...,y

= Z
sgn('''"Adx-®'"®dx\

rl x,...,u

\X'

' ' V/

by identifying

(

(X ' ' ' yx

dx^®- ' '®dx'',

(p is regarded as a section of 0^^ T'^iM), where the summation Y.K,...,V is taken over all the permutations of a , . . . , % and sgn("...7) denote the signature of permutation. A vector bundle G over M is called a suhhundle of F if the following conditions (i), (ii) are satisfied:

108

(i) (ii)

3. Differential Forms, Vector Bundles, Sheaves

For every peM, Gp is 3. linear subspace of Fp. Take a sufficiently fine locally finite open covering U = { Uj}. Then there exist /x C°° sections CTJI, ..., aj^ of F over each Uj, such that Wji(p), • • • 5 (^jfji(p)} forms a basis of Gp for any p e Uj.

In this case { o ) i , . . . , o}^} forms a basis of G over Uj, and, if we represent an element of G^ as Xa = i ^J^^jaip), (^j, • • •, ^j^) gives the fibre coordinates of G over L^. If vector bundles F and G over M are subbundles of a vector bundle //, and Hp = Fp + Gp for every point pe M, H is called the Whitney sum of F and G, and denoted by F®G. In this case if {e^i,..., e^^} and {yji,... ,/Jv} are bases of F and G over Uj respectively, {eju . • •, ^jfjL.fju • • • ,^1^} forms a basis of F © G over Uj. The set of all r-forms at each point of M forms a subbundle of 0^^ r * ( M ) , which we denote by r times

A T'^(M) = T'^iM) A • • • A r * ( M ) . (T) C°° sections dx/ A • • • A 6?x/, 1 ^ a < • • • < y ^ m, of ® ' r * ( M ) over Uj form a basis of A'^ T'^(M) over Uy. A C°° r-form on a domain W^ M is a C°° section of A ' T*(M) over W. For a complex vector bundle F over M with the transition functions fjk(p) = (fjki3(p)), F denotes a vector bundle over M with the transition functions fjk(p) = (fjk/3(p)), where ~ denotes the complex conjugation. Suppose that M is a complex manifold. Let { z j , . . . , z^,...}, Zj = (z],..., Zj) be a system of local complex coordinates of M Let T(M) be the tangent bundle of M, which consists of all tangent vectors of the form Sa t^7(5/5z") with v"eC. Then T(M) is the set of all tangent vectors of the form Xa ^fid/dzf) with t^/eC, which we denote by f ( M ) . The dual bundle r * ( M ) of T(M) consists of all (0, l)-forms X« %a dzj at each point of M, and the dual bundle f'^iM) of f{M) consists of all (0, l)-forms Sa 9jocdzJ. T{M) and T'^iM) are holomorphic vector bundles. Let S be the underlying difierentiable manifold of M, and Tc(2) the vector bundle over M consisting of all tangent vectors of 2 with complex coefficients if^^ u/Ca/ax/) with i;/ e C Since z/ = x,^""' + ix,^", we have

a = l

dXj

a=:.l

C^Z^-

a-^\

O^j

Hence we have rc(2) = T{M)@ T{M).

(3.58)

r2(i:)-r*(M)©f*(M).

(3.59)

Similarly we have

§3.3. Sheaves and Cohomology

109

A C°° section of the tensor product of arbitrary times of T{M), T'^iM), f{M) and f'iM) is a C°° tensor field on M g ) ' T{M) is a holomorphic vector bundle, and its holomorphic section is a holomorphic contravariant tensor field. Similar results hold for g)'' T'^iM). We denote by p times

q times

A T'^iM) A T*(M) = T'^iM) A • • • A T*(M) A T'^iM) A • • • A T * ( M ) the subbundle of A^ ^ c ( ^ ) , f = P~^^, which has p times

q times

{rfz; A • • • A Jzf A dzj A •'• • A t/z/l l ^ a < - - - < ) 8 ^ n ,

l^r<-"<5^n}

as a basis over L(,. Then a C°° (/?, ^)-form is a C°° section of A'' r * ( M ) A^ T*(M). A'' r * ( M ) is a holomorphic vector bundle, and its holomorphic section is a holomorphic ;?-form.

§3.3. Sheaves and Cohomology (a) Sheaves First let M be a differentiable manifold. By a local C°° function of M we mean a C°° function defined on a domain U<=^ M. We denote by D ( / ) the domain of / / is called a /oca/ C°° function at /? if /? e D{f). We say that two local C°° functions / and g at p are equivalent at p i f / = g on some neighbourhood of p. We denote it by writing / ~p g. It is obvious that ~p is an equivalence relation. By a germ of a C°° function at /? we mean an equivalence class of local functions at p. The equivalence class to which / belongs is called the germ off at p, and denoted by fp. The germ fp can be regarded as a C°° function defined on an "infinitely" small neighbourhood of p. Let s^p be the set of germs of all complex-valued C°° functions at p. Let <^, (A G ^p, and suppose that (p =fp and if/ = gp. Then we define a linear combination of cp and i/^ by putting Ci(p + C2(A = ( C i / + C2g)p,

Ci, C2 G C .

Clearly this definition is independent of the choice of / and g. Thus s^p becomes a vector space over C. Moreover we define the product of

which makes Mp a ring.

fp =
110

3. Differential Forms, Vector Bundles, Sheaves

Let ^ = U ^p peM

be the union of all ^p, pe M. We define a topology on sd as follows: For a point (pes^p^ s4, choose any local C°° function / with fp = cp, and an arbitrary opeii set U with pe U(=: D(f), and put ^{
{f,\qeU}.

Of course (p=fpe U((p;f, U). We introduce a topology on s^ by taking the family {n
=

cp,peUc^D(f)}

consisting of all ^{(p;f, U) w i t h / and U satisfying the above conditions as the system of neighbourhood of(pesi. Namely, a subset ^ c: j?^ is defined to be open if and only if U contains some %L{(p\f, U) for any cpeU. We denote by O the family of open sets of si. This definition makes sense. In fact, let ^ , r G C) and (p G ^ n T. Then there are neighbourhood ^{(p\f,U) and "Uicp'.g, V) such that ^{(p'J, U)^^ and ^{(p\g, V)^r. S i n c e / ^ ^ gp = 0, f{x) = u{x) and g(x) = v{x) on \x\ < 8. Hence for e G t/ n V, 0<s<8,f, = u, = v, = g,. Define m: M^M by m{sdp)—p. Then xcr satisfies the following conditions: (i) (ii) (iii)

TtT is a local homeomorphism, that is, for any (p e sd there exists a ^{
Proo/ (i) Since, restricted on ^{(p;f, U), m is represented by

§3.3. Sheaves and Cohomology

111

m maps certainly ^{(p;f, U) bijectively onto U. For any if/=fqe^{(p;f, U), a sufficiently small ^(il/',f, V), qe Vc: JJ, is mapped bijectively by m to the neighbourhood V of q. Hence tir is a local homeomorphism. (ii) is clear. (iii) Let Af = ^i<^ + (^2^ with m{(p) = m{ip)=p. It suffices to prove that, for any ^{x\L W, there are sufficiently small ^{cp; g, V) and ^((A; K ^ ) such that c^^+C2'Y]e%L{x'J,U) for ^e%i{(p\ g,V) and rj e ^{il/-, h, V) with m{^) = m(r]). Since Cigp + C2hp = c^cp -\-C2il/ = x =fp.

f coincides with Cig + C2h on some neighbourhood yc= U of p. Put ^ = tu-(^) = tJT(i7) G y. Then we have Ci^-^C2V = c,g^ + C2h^=f,enX',f,U).

I

s^ is called the sheaf of germs of C°^ functions over M. Let M be a complex manifold. Then using holomorphic functions instead of C°° functions, we obtain the definition of the sheaf of germs of holomorphic functions ^ = [JpeM^P- Let Zp = (zl,..,,Zp) be local complex coordinates centred at p of M. Then a local holomorphic function at /?, that is, a holomorphic function defined on some neighbourhood of p, has a convergent power series expansion: /=p(z,) = p(z;....,z;) 00

=

I

a.,..™„(4)'"' • • • (^r--

mj,...,m„=0

/ and g are equivalent if and only i f / and g have the same power series expansion P(zp). Thus a germ of holomorphic functions at p corresponds in one-to-one manner to a convergent power series in z ^ , . . . , Zp. Hence, by identifying a germ of holomorphic functions with the corresponding convergent power series, we may identify Op with the convergent power series ring C{z]„ . . . , Zp}: Op = C{zl,...,z;}.

(3.60)

If in the definition of U{(p\f U) = {fq\qe U), (peOp, U is assumed to be connected, U{(p;f U) is uniquely determined by cp and U. For, by Corollary to Theorem 1.7, / = g on U if fp = (p = gp since U is connected. Moreover if (p,il/e Op and ^ 7^ ^A? L / ( < ^ ; / t / ) n ^ ( ( A ; g , t/) = 0

112

3. Differential Forms, Vector Bundles, Sheaves

if U is connected. In fact, if/^ = gq at some qe U,f= g on all of U, hence (p = fp = gp = ip. Therefore the sheaf of germs of holomorphic functions 0 is Hausdorff. Define m: €-^ M by m(€p)=p. Then, as in the case of sd, m is a local homeomorphism, and the linear combination Ci(p-hC2il/, tjj(^) = tzT((/^), depends continuously on (p and il/. Let M be a differentiable or a complex manifold. In what follows, we denote by K one of C, IR, and Z. Definition 3.5. A topological space 5^ is called a sheaf OVQT M if the following conditions are satisfied: (i) (ii) (iii)

A local homeomorphism tjr of 5^ onto M is defined. For any pe M, vT~\p) is a X-module. For Cj, C2GX, the linear combination Cicp + C21I/, of (p^ipeS and XXT((P) = mill/), depends continuously on cp and ij/.

We call &^p = m~^{p) the stalk of S over /?, and tu- the projection of 5^. The above-mentioned M and ^ are typical examples of sheaves. We define the sheaf M"" of germs of C°° r-forms on a differentiable manifold M, and the sheaf O of germs of holomorphic vector fields on a complex manifold M similarly. More generally, given a complex vector bundle F over a differentiable manifold M, we define the sheaf of germs of all C°° sections of F as follows. By a local C^ section of F at /? we mean a C°° section of F defined in a neighbourhood of p. Two local C°° sections a and T are called equivalent if CJ = r in some neighbourhood of/?. We denote a --pT if a and r are equivalent. Then ~p is an equivalence relation. The equivalence class of local C°° sections is called a germ of C°° sections of F. The class to which a belongs is called the germ of a at /?, and denoted t>y o-p. We denote the set of all local C°° sections of F at p by Mp{F). Put si{F) = UpeM ^ p ( ^ ) - Then si{F) is a sheaf over M just as si. M{F) is called the sheaf of germs of C°° sections of F. For example, taking A ' r * ( M ) as F, we have ^ ( A ' T'^iM)) = si\ For a complex manifold M, ^A^ ^ ^(/yp T*(M) A^ T'^iM)) is the sheaf of germs of C°° (;?, ^)-forms on M. In case F is a holomorphic vector bundle over a complex manifold M, the sheaf ^ ( F ) = UpeM ^ P ( ^ ) of germs of holomorphic sections of F is defined similarly. For example 0 = (D'(T(M)) is the sheaf of germs of holomorphic vector fields on M, and ^(A'^ T^{M)) is the sheaf of germs of holomorphic r-forms on M. 0 plays an essential role in the theory of deformations of complex structures. By a locally constant function we mean a local C°° function/with/= c^C on D(f). The sheaf of germs of locally constant functions is identified with the direct product M x C with the topology defined by the system of open sets 0 = { l 7 x c | t / c z M , cG C}, where U is an open set of M. In fact if we identify a locally constant function / with its graph D(f) x c «= M xC, then

§3.3. Sheaves and Cohomology

113

its germ fp at pe D(f) is identified with (/?, c) e M x C , and U{fp\f, U) = {fq\qe U} with L / x c c i M x C . We denote the sheaf M x C of germs of locally constant functions simply by C. Similarly we denote by Z the sheaf M xZ of germs of Z-valued locally constant functions. Definition 3.6. Let 5^ be a sheaf over M, and W an arbitrary subset of M. A section a of 9" over W is a continuous map a: p^a(p) of W into 5^ with (T{p)e^p.

As an example, let us consider a section of the sheaf of germs of C°° functions over an open subset. Let W^ M bo an open subset and f a C°° function on W. Then clearly the map p-^fp,p^ W, defines a section ofM over W. Conversely, any section a: p^ o-(p) of si over W is given by some C°° function f on Was a(p)=fp. Proof Each o-(p) is the germ of some local C°° function g^^\ a(p) = (g^^^)p. Define / by putting f=f{p) = g^^\p). Then / is well defined. Since a is continuous, for a neighbourhood U(a(p);g^^^, U) of cr{p), there is a neighbourhood U(p)c: U of p such that for qe U(p), aiq) e U{a{p); g'^\ U) = {(g^^^),

\qeU},

which means that o-{q) = (g^^^)q. Hence f{q) = g^^\q). Thus for any peW, f coincides with a C°° function g^^^ on U(p). Hence / is C°°, and o-{p) =

Similarly for the sheaf 0 of germs of holomorphic functions over a complex manifold M, a section a of € over an open subset W corresponds in a one-to-one manner to a holomorphic function f defined on Wbya(p)=fp. In this sense we can identify a section of 0 over an open set with a holomorphic function defined there. Note, however, that for a section cr, (T(P) is a power series P(Zp) whereas /(/?) is a complex number for a holomorphic function / Similarly we can identify a section of si over an open set with a C°^ function defined there. Similar results hold for sd{F) or 0{F), where F denotes a complex vector bundle over a C°° manifold or a holomorphic vector bundle over a complex manifold. Namely, let 5 be a section of F over an open set W. Then by putting cr{p) = Sp, pe W, we obtain a one-to-one correspondence between the sections a of s4{F) over W and the sections s of F over W. Thus we may identify them. Similarly a section of €{F) over an open set may be regarded as a holomorphic section of F defined there. Let 5^ be a sheaf over M. For a section a: p^a(p) over an open set WciM, W=a(W) = {a(p)\pe W} is an open subset of ^, and a: W-^W is a homeomorphism.

114

3. Differential Forms, Vector Bundles, Sheaves

Proof. Since tix: S-^ M is locally homeomorphic, for any q^W, there is a neighbourhood T of o-{q) in &" such that tir: T ^ y=txT(T") is a homeomorphism. Here V is clearly a neighbourhood of q. Since (T is continuous, there is a neighbourhood U^ V of ^ such that ^ = a{U)<^ V. m(o-{p))=p implies that m(^)= U, and ^ is an open set of V since m is homeomorphic. Thus for any qeW, there is a neighbourhood U^ W such that ^ = a(U)<^ W is an open set of 5^. Hence W is open in 5^. Since trr: ^ ^ L/ is homeomorphic for each % m: W-^ W is a homeomorphism. Hence its inverse a: W^W is also a homeomorphism. I Conversely, an open set W which intersects with each ^p, pe W, at a single point o-(p), defines a section a over W = m(W). We denote by 0^ the identity of the abelian group ^p. Then the map /? -> Op is a section of 5^ over M. Proof. It suffices to show that for any qe M, p^Op is continuous in some neighbourhood L^(^) of q. Since tu is a local homeomorphism, for a sufficiently small U{q) there is a neighbourhood ^ of 0^ such that m: ^^ U(q) is homeomorphic. Consequently its inverse p-> a(p)e^ is continuous, hence by (iii) of Definition 3.5, p-^Op = aip) — o-{p) is continuous on U(q). I Thus p^Op is a section of 5^ over M, hence by the above argument, {Op \p e M} is an open subset of 5^. Let W be an arbitrary subset of M with U^ T^ 0 . We denote the restriction of the map p^Op to W, which is a section over W, by 0. For sections o", r of 5^ over W^ we define its linear combination by Cia-h C2T: p^

Cia(p)

+ C2T(P),

Ci, C2& K.

Then the set of all sections of 5^ over W forms a K-module. Definition 3.7. We denote by r ( W, ^ ) the X-module of all sections of ^ over W. The identity of the module r(W;5^) is the section Oip^Op, pe W. If r ( W, 5^) contains no element other than 0, we denote r ( W, &") = 0. If W is open, we see from the above argument that r ( W^ si) may be identified with the vector space of all C°° functions on W. Similarly r ( W, €), T{W,sd{F)) and Y{W,€{F)) are identified with the vector space of all holomorphic functions, all C°° sections of F over W and all holomorphic sections of F over W respectively. Let U ^ W, and a a section of ^ over W. Restricting a to U, we obtain a section of 5^ over U, which is called the restriction of cr to U, and denoted by rjjo: ryi a-^ r^cr is a homomorphism of r ( W, 5^) into r ( U, 5^).

§3.3. Sheaves and Cohomology

115

Suppose given a finite number of sections (TX e r ( W^, ^), A = 1 , 2 , . . . , K If i7 = n i ^ i ^ A ^ 0 , for any C;,eK, X = l,...,p, the map /?-> Zl=i ^\^x{p),P^ (7, is a section of 5^ over (7, which we denote by XI = i ^A^AI

C;,(T;,= X c^rucr^.

A=l

(3.61)

A-1

If n i = i ^A = 0 , we do not define XI = i <^AC^A- In what follows, if we write Z l = i CA<^A, we always assume that H I ^ i ^ A ^ 0(b) Cohomology Group Let M be a differentiable manifold, 5^ a sheaf over M, and U = {Ui,..., Uj,...} an arbitrary locally finite open covering of M. First we shall define the cohomology group of M with coefficients in 5^ with respect toU. A 0-cochain c^ with respect to It is a set 0 = {o}} of sections where (Tj e r ( Uj, 5^) for each j . A 1-cochain c^ = {ajj^} with respect to U is a set of sections CTJJ^ G r ( (7, n C4, ^ ) for all indices j , k with L^ n 14 T^ 0 such that f^jk — ~^kj' In general, a ^-cochain c^ = {0"^. ./c } with respect to 11 is a set of sections

for all (^+ l)-tuples of indices ko,...,kq with Uj^n- - - n Uj^^7^0 which are skew-symmetric in the indices /CQ, . . . , /c^. Given two ^-cochains {o}.../}, {T^.../}, we define their linear combination by Ci{(Tj...i}+C2{rj...i} = {c^(jj...i-hc2Tj...i}, with Ci, C2GK Then the set of all q-cochains with respect to U becomes a K-module, which we denote by C^((7, 5^) or simply, by C^(U). For a 0-cochain c^' = {o}}, the 1-cochain c^ = {7,7,} defined by TJ^ = o-f,- aj is called the cohoundary of CQ and denoted by 8c^: ^{o-j} = {rjk},

Tjk = o-j,- o-j.

Note that by cTk-o-j we mean the linear combination of them in the sense of (3.61), hence by so writing, we assume also Ujn 1/^7^0- For a 1-cochain {0}^}, we define its coboundary by ^{o-jk}

= {Tjkl},

Tjk = O-ki - OTji + O-jk = O-kl + (Tij + C7);

>•

116

3. Differential Forms, Vector Bundles, Sheaves

Clearly T,^/ is skew-symmetric in 7,/c,/. In general for a (^-l)-cochain {o"/ci- fcj with respect to U, we define its coboundary by ^{o-kr-kj = {^/co-kj, ^ko--kq=

where

Z (-l)^Or^---/c,_i/c,+i---k^ s=l

= i (-l)X...x,..fe,,

(3.62)

s=l

where / means that we omit this index. 5 is a homomorphism of C^~^(U) into C^(U). If {T;^...fcJ = 5{(Tfc,...fcJ, then we have 5{T;^...fcJ = 0. In fact

q+l

/ ^~^

^"•"^

5=0

\r=0

/=s+l

= Z (-ir

I (-1)^...;,,..;,,..+ I

(-l)'-V...;,,..;,,..

\ /

Hence we have 55 = 0.

(3.63)

If 5c^ = 0, we call c^ e C^(U, 5^) a q-cocycle with respect to U. We denote the set of ^-cocycles with respect to U by Z^(ll, 5^) or Z^(U). Z^(U, 5^) = {c^ G C^(U, 5^) 15c^ = 0} is a submodule of C^(U, 5^). For q^l, SC^-^U, ^ ) c= Z^(U, 5^). We put

SdC^-'iVi, 5^) = 0 by (3.63), hence

/f ^(U, ^ ) = Z^(U, 5^)/6C^"HU, ^ ) ,

^ ^ 1,

which we call the cohomology group with coefficients in 5^ with respect to U. For ^ = 0, we define

If {cTjle Z^(Vi, 5^), then o} = cr^ on Uj n 11^9^0, hence it defines an element (T € r ( M , 6^) with a = CTJ on Uj. We have then CT, = r^cr. Thus we may identify Z\Vi, 5^) with r(M, 5^), that is, if^(U,5^) = r ( M , ^ ) .

(3.64)

§3.3. Sheaves and Cohomology

117

Thus we define /f ^(U, 5^) for any locally finite open covering U of M. //°(U, 5^) = r(U, 5^) does not depend on the choice of U. For q^\, however, f/^(U, 5^) does depend on U. Therefore we define the ^th cohomology group W ( M , 5^) of M with coefficients in 5^ as the "limit" of //'^(U, ^ ) , //^(M,5^) = l i m / / ^ ( U , ^ ) as each Uj of U becomes "infinitely small". The limiting process will be explained now. We say that the open covering 9S = { V;^} of M is a refinement of U if each y^ is contained in some member t 4 of U. We write 33 < U or U > 93 if 9S is a refinement of U. For each V),, we fix an arbitrary Uj^ with V^ cz C4^ and write it as ^/C(A)' K ^ ^/C(A)- We define a homomorphism of C^(U) into

ng:{a^....J^{T,^....J

(3.65)

by putting TAO-A,

= 'vo-fc(;,^)...fc(;,^),

where

V = Pi K, ^ 0 -

Since it is easily verified that

n g maps Z^(U) into Z^(9S) and 5C^~HU) into dC^-'i^), Hence n g defines a homomorphism of H^(U, ^ ) = Z^(U)/5C^"HlI) into //^(9S,5^) = Z^(9S)/5C^"HSS), which is also denoted by n g . n g : C ^ ( l l ) ^ C^(9S) depend on the choice of Uk(\)^Vx for each V^, but Ilg:/f^(U, 5^)^ //^(9S, 5^) turns out to be independent of this choice. In fact we have the following Lemma 3.2. The homomorphism n g : / / ^ ( U , y ) ^ / / ^ ( 9 S , 5^) is uniquely determined by U and 9S. Proofi For each A choose an arbitrary7(A) with Uj(^x)^ ^K, and put rix,-K, = ^vO}-(Ao)-7(A,), where

V = C] V^. ^ 0 .

It suffices to show that for {o-fc^...fcjGZ^(U), we have {r?.,-A,}-K-A,}eSC«->(«).

118

3. Differential Forms, Vector Bundles, Sheaves

Put

where VV^ = n L i K, 7^ 0 - Though in the indices, we define

(5'<)AO-A, =

K^^...^^

is not necessarily skew-symmetric

Z (-1)'/<:;,^...A,...A,. 5=0

Then we see by an easy calculation that ^Ao-A, - TAO-A, = ( 5 ' < ) A O - A , .

(3.66)

In fact if we write h^ for /c(Aj and js for 7(Aj, we have

(5'<:)AO-A, =

I s=0

(-l)''
= i i-Mi s=0

(-i)''-v<^v'.j.-/. v,

\l=0

r=s+l

f=0

/

\s=0

5=r+i

/

On the other hand by the hypothesis, we have, s= t

Hence we have

Thus (3.66) holds.

q


t=0

t=0

119

§3.3. Sheaves a n d C o h o m o l o g y

The above K^^...^ is not necessarily skew-symmetric in ^ i , . . . , ^q, so we put K.,....,=-Zsgn

A^r-Mq'

\/^i • • • Mg/

where the summation is taken over all the permutations i^i.'-^") of ^ i , . . . , Vq. Then {K.,....j€C''-HSS). By (3.66) V\o---\q

'^Ao-'-Aq

A.V

:Isgn

(g+1)!

(g+l)!

K^^)^'0•^^-, \^0

j\-::;;^)i(-i)x........

\j^o

_ J _ V

(^0

(^+1)!

\j^o

As • • • A A

^

I I (-1) f<.o-^s--,-

Since X^ (-l)^^^.^...^^...^^ is skew-symmetric in VQ, • • -, ^q, we have

''?Ao---Ag ~'7'Ao---Aq =

Z

(~"1)'^i'o•••A•••''q•

namely, {T/.,...A J - {T......J = 6{K.,....J G SC^-HSS).

We define U < 93 if 9S is a refinement of U. Then < defines a partial order on the set © = {U, 9S,...} of all locally finite open coverings of M, which makes © a directed set since for any U, 93 e ©, there is a 2B G © such that 93 < SB and U < SB. Write //^(U) for //^(U, 5) for simplicity. Then by Lemma 3.2, for U, S3 G S' with U > 93, a homomorphism ng://^(U)^H^(93) is defined. Clearly 11^ is the identity, and, if U > 93 > SB, we have (3.67) Now we define limu /f^(lt) as follows. First we define g > 0 for g e //^(U) i/ r/iere exists 93 < U , 93 G © , 5MC/I ^/lar n g g = 0. Put iV^(U) = {gG/f^(U)|g>0}, which is a subgroup of //^(U). In fact if n g g = 0, and

120

3. Differential Forms, Vector Bundles, Sheaves

U^h = 0, then taking X with 93 > 3E and U > X, we obtain by (3.67),

Put H^(U, ^ ) = ^^(U) = //'^(U)/AT^CU), and let n " : g ^ g be the canonical homomorphism of H^(U) = H^{U, 5^) onto H^(U,5^). For U > « , we have U^N^U) a iV^(9S). In fact for g e iV^(U), there is a SB< U with n ^ g = 0. For X e © with 9S <3£ and U < X, we have

nIn§g = nig = nfn^g = o, hence n§gGN^(9S). Consequently n § : i/^(U)^/f^(9S) /iomomorp/».sm n § ; H^(U) = H^(U)/N^(U)^H'^CSS) =

zWwces r/ie

H'^m/N'^m.

This fig is an injection. For if n § g = 0 with g = n"g, g e / f ^ ( l t ) , then nggGN^(9S), hence there is a SB< 93 with n i n g g = 0. Hence U^g = 0, which means gEiV^(U), that is g = 0. Thus for U < ^ , /f^(ll) maj; 6e identified with a submodule n § / / ^ ( l t ) i;ia r/ie inclusion n § : H^(U, 5^)^ H ^ ( « , 5^),

U > SS.

(3.68)

Taking the union [_)^^^ H'^(Vi, ^) for U G © , and considering it as limu /f ^(U, 5^), we define t/ie cohomology group ofM with coefficients in 5^ by H''{M,^)=

U H^(U,5^).

(3.69)

For any U G ©, n " : ff ^(U, ^ ) ^ H^(U, 5^) c: //^(M, ^ ) is a homomorphism of H^(U, ^ ) into //^(M, 5^). A g-cocycle C ^ G Z ^ ( U , 5^) defines an element g G H^(U, ^ ) = Z^(U, 5^)/5C^-HU, ^ ) , which in turn defines an element g = n^gG i/^(M, 5^). We call g r/ie cohomology class of a q-cocycle c^, and denote it by [c^]: [c^] = n"gG //^(M,5^). It is clear from (3.64) that H\M,^) = T{M,&'), For ^ = 1, we have the following

(3.70)

§3.3. Sheaves and Cohomology

121

Theorem 3.4. n " : H\U, ^)^H\M,

5^) is injective.

Proof. First we shall prove that N\Vi) = 0. Let geN\Vi). Then there is 9S 2B, hence n ^ g = n ^ n ^ g = O. Here by Lemma 3.2, we may assume that H ^ is defined via the inclusion Wj^ ^ Uj. Suppose that g is represented by a 1-cocycle {o}fc}eZ^(U). If we define U^: Z\n)^Z\m) via the inclusion Wj^ c ^ , then n§5{(7,7c} = {rjxkv},

Tj^kv = fwj,r.Wj,^o-jk,

with

WJ;, n IK^ ^ 0 .

Since n i g = 0, {rj,k.} e 5C'(SB), hence '^A kp ~ '^kv ~ "^jX

for some {T^AI ^ C^(S[B). Since ajj = 0, r^^ - TJ^ = TJ^J^ = 0, which implies that Tj^ = Tjx on Wj), n WJv 7^ 0 . Therefore we define a section r^ e r ( Uj, &") by putting Tj = Tj^ on ^ n W, = W;A. On ^ n L^^ n W^, o^-fc = T,-A/CA = Tfc - TJ,

hence o}^ = T' /'C - '^j on Uj n L^^, that is, {cr^/,} = Thus we obtain iV^(U) = 0 , which impHes

5{T^} G 5 C ^ ( U ) .

Hence g = 0.

H\Vi,^) = H\U,Sf). Therefore H " : / / H U , ^ ) -> H H U , ^ ) c: H\M, &") is injective. Corollary.//HM,^) = U u e © ^ H H , n

(3.71) I

I

Theorem 3.5. / / / / ' ( U , , ^ ) = 0 /or 6i;er>; ^ G U, r/?en if ^U, 5^) - H\M, ^). Proof. By the Corollary above, we have H\M, 9") = {Js8^^ H\SS), hence it suffices to prove H\^) a H\n) for any 9S G ©. Let U = {Uj} and 95 = { VJ, and put SB = { W);, | W,;, = Uj nV^,^ 0 } . Then by (3.71) and (3.68), H\SQ) a H\S&), hence we have only to show that //HSB) ci H\Vi). For this purpose it suffices to prove that for any 1-cocycle {r^;^ kv}^ Z^SB), there are a 1-cocycle {cTjk}eZ\Vi) and a 0-cochain {T^JG C^(2B) such that 'TJAfc.^= ^Wj^r^ W^yjk

+ '^fci. - TjX'

(3.72)

In fact (3.72) means that {r^AkJ = ni{o}fc} + 5{'7}A}, which implies //^SB) c H\U). For a fixed L(j, 2B^ = {WJx} is a locally finite open covering of Uj. Since by hypothesis H\UJ, &>) = 0, we have Z\mj)/SC°i'SSj) = H'(2B,, 5^) = 0 by Theorem 3.4. Hence {r.vvfcj e Z'(SB^) = 5C°(2B,), i.e., there is a {T,^} e €"(28;)

122

3. Differential Forms, Vector Bundles, Sheaves

such that

Fixing such

{TJ^,}

for each Uj, and putting ^j\ kv = ^j\ kv - Tku + Tj\,

(3.73)

we have {(7,xfcJeZH2B), and C7,A> = 0. Since o^A^v + ^^vk^x + o-fc^iA = 0 , and o}Afc^ = -o-k^jx, we have cT^A/cy^ = ^>/CA^- Similarly we have o^Ak^ = ^jAfc^- Hence by putting o-j^^ = crj;,f^^ on U, n L^^ n V^ n V^ ^^ 0 , we define a section o)/,^ G r ( Uj nUj^n V^). a}Ak^ = o-j^ku implies that o^-fc^ = o}^, on LTy n t/^ n V^ n V^ 7^ 0 . Hence again by putting cr^fc = (7,^/0^ on L/jnU^nV^y^ 0 , we obtain a section a}fcGr(L^n t 4 ) . {o}/,} is a 1-cocycle since O-jk + O'fc/ + (^Ij = O-jx kk + CTfcA /A + 0-/AJA = 0

on each Ujn UknUir)V^9^0. (3.73). I Theorem 3.6. / / H\Uj,^) H^{M, &^) is injective.

Then (3.72) follows immediately from

= 0 for each

UJEU,

then

U^:H\n,5e)->

Proof. It suffices to prove N\n) = 0. Let g e iV^(U), then there is a 93 < U such that n g g = 0. Let n = {Uj} and SS = {K} and put SB^ { V^A I W;A = Uj n VA ^ 0 } . Then ni^g = 0. Suppose that g is represented by a 2-cocycle {(7,^^} G Z ^ ( U ) . If we define n ^ : Z^(U) -> Z^(SB) via the inclusion Wjx ^ Uj, we have ^'SsWijkf

— {'^iXj/JL kA^

'^iA7> kv =

^ W , A O W ^ ^ O Wfc^^yfc-

Since n ^ g = 0, there is a 1-cochain {r/Afc^Je C^SB) such that '^ikjtJLkv = '0>fc»' "^ '^fc^'A + '"iA j>-

(3.74)

Since for each 7, o};^ = 0, we have '^7> 7> " ^ T/> JA "•" '^A j^x ~

'^jK j>i, jv — ^ j

which means {r^A^leZHSB^) with 2BJ = {S[B^A}- SB^ is a locally finite open covering of Uj. Since ff H ^ , ^ ) = 0, there is a 0-chain {r.^JG C^(SB^) such that jfA jV

v*^

\/A*

§3.3. Sheaves and Cohomology

For each Uj fix such a

123

{T,A},

and put

^j\ kv ~ '^jk kv ~ '^kv ' '^Aj

then by (3.74) we have '7"iA jfjL kv — ^jyL kv'^

^kvi\'^

^ikjix'

(3.75)

Putting f=7, we have T,A7>/CI. = 0 since o-jj}, = 0. On the other hand, since o-jkj^=0, we have o^xk^ = co>fc^- Similarly we have o-j^j,^ = aj^j,^. Hence as in the proof of Theorem 3.5, we define a 0-cochain {a)^} e C^(U) by putting cfjk = (^j\kp on Ujn UknV;,nV^9^ 0 . Putting IJL = P = X, by (3.75) we see that cTijj^ = (Tjj^ 4- o-fcj + cT-y on each t/,- n UJD Uj^n V^. Hence {o-y^} = 6{o}fc} e C^(ll), which means g = 0. I (c) Exact Sequence Let 5^ be a sheaf over a differentiable manifold M, and XJT: 5^->M its projection. Definition 3.8. A subset 5^' c: 5^ is called a subsheaf of 5^ if the following conditions are satisfied. (i) (ii)

5^' is an open subset of 5^. ti7(5^') = M.

(iii) For any peM,

5^^ = rxT~\p) n 5^' is a X-submodule of 5^^.

Clearly 5^' itself is a sheaf over M Let 5^" be a sheaf over M with the projection m". Definition 3.9. A homomorphism h of ^ into ^" is defined to be a continuous map of 5^ into ^" satisfying the following conditions: (i) (ii)

m" ° h = m. For any pe M, h: ^p^ ^p is a. iC-homomorphism.

From (i) it follows that h{9'p)(^^p. Since m and TZT" are local homeomorphisms, h is also a local homeomorphism by (i). Let /i: 5^-^ 5^" be a homomorphism. By h((p) = 0,

124

3. Differential Forms, Vector Bundles, Sheaves

/z(5^) is a subsheaf of 5^". In fact, /z(5^) is open in 5^" since /i is a local homomorphism. The other conditions are obviously satisfied. A homomorphism h: 6^ -» 5^" is injective if and only if ker h = {Op \pe M}, which we denote by ker h = 0. Definition 3.10. h: 5^^5^" is called an isomorphism of 9' onto 9"' if /i is a homomorphism of 5^ onto 9" and if, for any pe M, h: 9p-^ 9p is a Kisomorphism of 5^^ onto 5^^. If there exists an isomorphism of 5^ onto 5^", 9 and 5^" are said to be isomorphic, and is denoted by 5^ = 9". We often identify 9 with 9" if 9 = 9". Since a homomorphism h: 9^9" is a local homeomorphism, h: 9^ h{9) is an isomorphism if ker h — 0. In this case if we identify h{9)c: 9" with 9 via /i, then h can be regarded as the identity 9-^9^ 9". Thus being regarded, h is called the inclusion map, and often denoted by i. Let 5^'c: ^ be a subsheaf of 9. Let ^^ = 9p/9'p be the quotient group, and let 2. = U^eM ^P- Define tzr: ^ ^ M by m{^p)=p. Let /i^ be the natural map of 9p onto 1^ = 9p/9'p, and define /i: 9^ ^ by putting /i(
= m{il/)}

is a neighbourhood of a + j8. For, if we put U = m{%) = m(T), for sufficiently small %V, m'.^^U, and tu: T-> L'^ are both homeomorphisms. Their inverses a: p^ (p = cr(p) and r: p^ il/ = r{p) are sections of 5^ over U. Hence O'+T: P^ (p + ilf = o-{p)^-T{p) is also a section of 5^ over U. Hence ^-\-Y = {o-{p)-\-T{p)\pe U) is open. Since by definition 9' is an open subset of 9, this implies that for any open Wci9, h-\h(W))

= {(p + ilj\(peW,il/eW,xu((p)

= m(ilj)}

is open. Thus h(W) is open in % that is, h is an open map. For any point h(a), a e 9, of 2., take a neighbourhood ^ of a such that m: ^^ U is Si homeomorphism. Then by the above argument / i ( ^ ) is an open subset of <S. Moreover for any open subset T^% h{T) is an open subset of /i ( ^ ) . Conversely, if f is an open subset of /z ( ^ ) , r = /i" H ^ ) n ^ is an open subset of % and h{T) = T. Hence h: ^ ^ / i ( ^ ) i s a homeomorphism. Therefore h\ 9^91 is a local homeomorphism. Since tzr: ^-> L^ is a homeomorphism, and 7r° h = xjj, m: / i ( ^ ) - » ^ is also a homeomorphism. Hence TT: ^ ^ M is a local homeomorphism. The continuity of Ci(p-\-C2^ with 7r(^) = iriiji) with respect to cp, ij/e^, follows from the continuity of Ci(p-i- C2il/ for (p,il/e9. Hence <S is a sheaf over M with the projection TT. I

§3.3. Sheaves and Cohomology

125

Definition 3.11. ^ is called the quotient sheaf of 5^ by 5^' and is denoted by ^/9". /i: 5 ^ ^ ^ is a surjective homomorphism of 5^ onto ^. For a homomorphism h'.^^^\ put 5^' = ker/z. Then /i maps 5^/5^' isomorphically onto h{^), hence h{^) = Sf/9". A homomorphism h: 5^^ 5^" of sheaves over M defines homomorphisms of cohomology groups H"^{M, ^) ^ H"^{M, Sf") as follows: First for a section a: p-> o-(p) of S over a subset W^c M, ho".

p^h(a(p))

is a section of 5^" over U^, and /i: or-> /icr is a X-homomorphism of r ( W, 5^) into r ( W, 5^"). Next let U = { Uj} be a locally finite open covering of M. For any ^-cochain c^ = {0-;^...^ }G C^(U, 5^), by defining hc"^ = {hai^...i^}, we obtain a homomorphism hic^^hc'^ of C^(U, ^ ) into C^(U, ^ " ) / s i n c e h° 8 = 8 ° h follows immediately from the definition (3.62) of the coboundary, h maps Z^(U, 5^) into Z^(U, 5^") and 5C^"HlI, ^) into 5C^~HU, '^'O, hence h induces a homomorphism

Let SS = { V;^} be an arbitrary refinement of U = { Uj}: U > 93. For any V^, choose an arbitrary member Uk(x) of U with L^fc(A)=^K, and define n g : C^(U, 5^) ^ C^(9S, 5^) and n g : C^(U, 5^") ^ C^(9S, 5^") by (3.65). Then clearly hoU = Uoh holds. Hence also for n g : //^(U, 5^) ^ /f ^(93, 5^) and n g : //^(U, 5^")-^ ^^(SS, 5^"), we have

/iong = ngo/z. Therefore if n g g = 0 for g G / / ^ ( U , ^ ) , then ng/ig = 0. Thus g > 0 implies that hg>0, that is, /i maps iV^(U, ^ ) = {g6//^(U, ^ ) | g > 0 } into N^n, T) = {g"G //^(U, 5^'01 g"> 0}. Consequently h induces a homomorphism /2u: J?^(U,5^)^J^^(U,5^"). By (3.68) if U>9S, we have H^(U, 5^)c: ^^(SS, 5^), and H^n,Se')cz W(S8^ 9^"). It _ follows immediately from _ /i o n g = n g o /i ' that hss: H\S8^ ^ ) -> /f ^^(95, ^") coincides with h^ on //^(U, ^ ) . Since

by putting /i = /i^ on each H^(U, 5^), we obtain a homomorphism h:H^M,9)-^H^M,9").

126

3. Differential Forms, Vector Bundles, Sheaves

We write ^ - ^ T for a homomorphism /i: ^ ^ ^". Definition 3.12. A sequence of sheaves and sheaf homomorphisms over M ^0 - ^

> ^m-i -^^^^^^ ^ . - ^

> ^/

(3.76)

is said to be exact if /im-i('^m-i) = ker/i^,

m= l,2,...,/-1.

Similarly a sequence of i^-modules and i^-homomorphisms Ho

> . . . —> H ^ _ j

> //^

>. •.

is said to be exact if hm-i(Hm-i) = ker /i^,

m = 1, 2 , . . . .

Suppose given an exact sequence of sheaves over M 0 -> ^ ' -^ ^ -^ 5^" -» 0.

(3.77)

We investigate the relation between the cohomology groups H^(M, 5^'), H^M, ^), and H^(M, ^'O- The exactness of (3.77) implies that ker t = 0, ker h = L{&"), and S"'=h{Sf). Hence ^ ' = 6(^)c: ^ is a subsheaf and t is the inclusion map. Hence also ^" = &"/&". We write H^(5^') for /f^(M, SF') etc., for simplicity. The homomorphism 5^' -^ 5^ induces a homomorphism H\^') -^ //^(5^), and 5^-^ ^" induces /f^(5^) -^ H^(5^"). The following theorem is fundamental. Theorem 3.7. The exact sequence of sheaves 0-» 5^' -^ 5^-^ 5^" —> 0 induces the exact sequence of cohomology groups 0 ^ H\^')

-4 H\^)

-X H\5e') ^ H\5e') A . . .

i ; H\5e') A //^(5^) -4 H^(5^") ^ //^•^H'^O -^ • • •.

(3.78)

Before we give the proof, we explain how 5* is defined. First for ^ = 0, let us consider how 5* must be defined in order that the sequence /f 0(5^) J^ H%^") -^ H\^')

(3.79)

may be exact. The exactness of (3.79) means that for a given o"" E H^i^Sf"^ — r ( M , &"'), there is a ae H\9') = r ( M , ^ ) suc/i that ha = a" if and only if 5*0-" = 0. Since the projections vT,xn" of Sf, if" are local homeomorph-

§3.3. Sheaves and Cohomology

127

isms, and h is also a local homeomorphism, for a given cr'\ if we take a sufficiently fine locally finite open covering U = m } , there is a section (Tj e r ( Uj, Sf) for each J such that hdj = rjj.cr". By (3.64), the 0-cocycle {r^o-"} gives the section a". Hence if we put c^ = {cr,}e C^(U, 5^), /ic^ = {/zo}} = {ru.(T"} = (T". Thus for a"eH\Vi,S"'), there exists a c^eC^(U,5^) with he = a". Note that c^ may not be a cocycle. Take any such c^, and put 6c^ = {r'jk\^ TJfc = o-k~ ^r Since /irj^ = /lo-fc — /lo) = 0, and ker h = 5^', we see that TJfcGr(U,n t 4 , ^ ' ) . Since 55c° = 0, 5 C ° G Z H U , ^ ' ) . We define 6*C7-"G H\M, ^') as the eohomology elass of the I-cocycle 8c^: 5*c7" = [5c^]. Since by Theorem 3.4, H\U, &")^H\M, ^ ' ) , we may consider 5*0-" = [5c^] e f/HU, '^') = Z\Vi, ^')/8C\Vi,

&"),

5*0-" is determined uniquely by a". For, if cr" = 0, then /ia} = 0, hence c^ = {cr,} G C°(U, 9"). 8"^a" is also independent of the choice of an open covering. If there is a cr G r ( M , 5^) with ha = cr", we have 5*o"" = 0. In fact, if we put c^ = {rij.cr}, then hc^=o-", hence 5c^ = 0. Conversely, assume 5*0-" = 0. Then 8c^ = {r^,,}e 8C%n,^"), that is, or^ - cr, = rj^ = o-^, - o-j with crJG r(U, 5^')- For UjnUkT^ 0 , CT) - o-j = o-^ - o-^, hence putting o- = aj- a] on each tT;, we obtain a section cr G r(M, 5^). Since /icrj = 0 and ho-j = ru.a", we have ha = a". Hence (3.79) is exact. Now we assume q^l. We must define 6* so that //^(^) -i H%^") ^ H'^^^Sf')

(3.80)

may be exact. Let M be a complex manifold. Suppose given for every p a. local complex coordinate system Zp = ( z ^ , . . . , z^) centred at p, and a coordinate polydisk

Usip)(p) = {zp 1141 < ^(p)^ '",K\<

^(p)h

^(p) > 0.

Then we may choose, by Theorem 2.1, at most countably many coordinate polydisks U^ with ^A=^e,(/?A)^^.(p.)(PA),

0<S,^£(pJ,

A= 1,2,...,

such that U ={Ux} is a locally finite open covering of M. Similarly for a differentiable manifold, using local C°° coordinates ( x ^ , . . . , x^) centred at p and an open set Us(p) = {Xp I |x^| < £ , . . . , Ix^'l < s}

with e > 0,

we obtain the same result. Here we assume that the closure [U^(p)] is a compact subset of the domain of local C°° coordinates (Xp,..., x^). We call Usip) a coordinate (multi)interval of radius e. For a complex manifold

128

3. Differential Forms, Vector Bundles, Sheaves

M, too, we may consider coordinate multi-intervals, regarding M as a differentiable manifold as usual. Let U = { Uj} be a locally finite open covering of M, and suppose [ Uj] is compact. Then we may choose an open subset Wj of each Uj such that [l^.Jc: Uj and that SB = {W;} is already a covering of M. Proof. Since for peM, there are only a finite number of Uj with pe Uj, taking s(p)>0 sufficiently small, we may assume that [ U^(p)(p)] c= Uj if p e Uj. Choose such Ue(p){p) for each pe M, and take

so that {VA IA = 1, 2,...} is a locally finite open covering of M. Then px belongs to one of Uj, and if px e Uj, then [Vx'\^[U^(p^)ipx)]^ Uj. Thus for each A, [V;^] is contained in some L^: [ V^]^ Uj. Then if we put Wj=

U

v„

we have U j ^ = U^ VJ = M, hence SB = { W)} forms an open covering of M. Since [L/,] is compact, and {V^} is locally finite, there are only a finite number of V, with [ VJcz C/,.. Hence [ Wj] = Ufv,]c=./, [ ^ A ] ^ ^ . I In what follows we denote by U, SS, SB,... locally finite open coverings of M. Lemma 3.3. Given a q-chain c"^ e C^(U, 5^"), we can find a locally finite refinement ^ of U and c^ e C^(aS, 5^) ^MC/I r/iar ngc"^ = he''. Proof. Put U = { L^}. We give proof only for the case q = 2, but the generalization is straightforward. Since for ^ < SB< U, E ^ = I l ' ^ n ^ , replacing U by its appropriate refinement if necessary, we may assume that the closure [ Uj] of each Uj is compact. Then there is an open covering SB = {W}} with [Wj]czUj.

First let c"^ = {a'lj,,} e C\n, T), a'ljj, e r ( ^ n Uj n Uj,, &"'). Take an arbitrary point peM, and denote by U^ip) the coordinate multi-interval of radius e > 0 with centre p. Suppose pe Uin Ujn U^, then since m, m" and h are local homeomorphisms, for a sufficiently small U^{p)^ Uin Ujn Uk, there is an element TeT(Ueip), ^) such that hT= rjj^f^p^^a'ljk. Since 11 = {Uj} is locally finite, there are only a finite number of Ui n Ujn U^ with pe UinUjn U^ for a fixed p. Hence we can find for each p a sufficiently small U^ip) satisfying the following conditions: (i) (ii) (iii)

If p G Ui n UjD Uk, U^{p)cz Ui n Ujn U^, and there is a r e r ( / 7 , ( p ) , 5^) with r^^(^)cr;;.fc =/IT. If i? G Wj, then U,{p)c:Wj. l f [ \ ^ ] n ^ , ( p ) ^ 0 , t h e n r/,(/7)c=^.

§3.3. Sheaves and Cohomology

129

In fact (ii) follows from the local finiteness of 2B = { WJ}. We can verify (iii) as follows. Since for a sufficiently small U^^ip), there are only a finite number of Uj with U^J^p)n Uj9^0, we can choose a sufficiently small s with 0<s<so such that for pe Uj, U^(p)<= Uj, and that for p^ Uj,[ Wj]n U,{p) = 0. Note that here we have [Wj]<^ Uj. We fix U^ip) = Ueip)ip) satisfying (i), (ii) and (iii) above for each pe M. Choose coordinate multi-intervals Vx^U^SP^^'^^eipJPx),

0<8,^e(;7j,

A = l,2,...,

such that 93 = {VA} is a locally finite open covering of M By the condition (ii), each V^ is contained in some WJ. For each A, choose an arbitrary WJ(A) with VA ^ Wjt^xy Then V^ c: Wj^^^^cz Uj^^.^, hence 93
V= V^^n

V^nV,9^0.

We must prove the existence of T;^^^ e r ( V^ ^ VJ^ (^ K, '^) such that /ITA^^ = T^V for each r^;^,. Put i=j(X),j=j{^) and /c=7(^). Then V,nW,^0 since V^ <= W,, V^ c: WJ., V^ c= W^ and V^ n V^ n V^ T^ 0 . Therefore we have

WjnU,^,^^(p,)^WjnV,9^0, hence by (iii), U^(p^)^ Uj. Similarly we have U^i^p^^ip^)^ U^, while by (ii), ^e(p^)(P\)^ ^ ^ ^i' Hence we have V,c:U^^,Jp,)uU,nUjnUj,. Consequently by (i), there is a TeT(V^, 5^) with hr = ryyijk. Put TA^,. = ryr with V = V^ ^ V]^ (^ K- Then we have

as desired.

I

For a sheaf 5^, we denote the cohomology class with respect to U of a ^-cocycle C ^ G Z ^ ( U , 5 ^ ) by

[cju G //^(U, ^ ) = Z^(U, 5^)/5C^-HU, 5^).

130

3. Differential Forms, Vector Bundles, Sheaves

Then the cohomology class [c'] of c ' is given by [c«] = n " [ c « ] u e / f « ( M , ^ ) . For U>SS, n § : / f « ( U , ^ ) ^ / / ' ' ( S S , 5 ^ ) induces an inclusion H^iSQ, &>). Hence

H''{n,9')-^

n"[c'']„ = n*ng[c'']u, hence in view of n§[c^]u = [n^c^]ig, we see that [c''] = [ngc^],

(3.81)

which means that the cohomology class of c"^ does not depend on the choice of refinements of a locally finite covering. Lemma 3.4. For any g"e H^{M, 5^")> by ^ suitable choice of U, we can find a q-cocycle c"^ and a q-cochain c^ such that g" = [c"%

hc'' = c"^eZ%n,^"),

and

C^GC^(U,5^).

(3.82)

Proof Let g" be given as g" = [6"^] by some g-cocycle 5"^ e Z^"^, ^"). By Lemma 3.3, taking a suitable refinement U < 93, we can find c^ G C^(lt, 5^) such that he"" = litb'"'. Put c"^ = ng6"^ G Z ^ ( U , Sf"). Then by (3.81), we see that (3.82) holds. I For any g"eW{M,^"), Lemma 3.4. Then since

take

55c^ = 0,

C"^GZ^(U,

5^") and

C^GC^(U,^)

as in

hdc"" = 8hc^ = 8c"'' = 0,

and ker h = S\ we have Sc" e Z^-^^U, 5^'). We define 5*g" by 6*g'' = [5c^] G W^\M,

^').

(3.83)

Then 5*g" /s determined uniquely by g" and independent of the choice of U, c"\ and c^ used above. Proof Take another b'"" e Z""(SQ^ S") and 5^G C^(9S,^) such that g" = [6"^] and hb^ = b'"'. We must show [6fe^] = [5c^] in H^^\M,5e'). Take an arbitrary SB with 2B
hc^ = c"\

hb"" = b"\

§3.3. Sheaves and Cohomology

131

If we put a" = h'"^ - c"^, and a = b'^ - c^, then we have [a"] = 0,

ha = a"6 Z^(SB, ^ " ) ,

a e C^(S[g, ^ ) .

We must prove that [8a] = 0 in H^'^\M, ^ ' ) . Since [a"] = 0, taking a suitable 3E<SB, we can find e"6 C^-'(X, ^") such that n f a"= 6e". By Lemma 3.3, we can find e e C ^ ' ^ O , 5^) such that he = U^e'\ where g is a suitable refinement of 3£. Since n | ^ = n | n | ^ , we have hUfa - Ufa" = U^Se" = dU^e" = 8he = h8e, hence putting a' = lifa-8e, a ' e C ^ O , 5^0, hence

we have ha' = 0. Since ker/i = 5^', we have

n f 6a = 8Ufa = 8a'e C ^ O , 5^'). Therefore [6a] = [ n f 6a] = [8a'] = 0 in H''^\M, &"). I Next we shall prove the exactness of (3.80). We must prove that for g"eH^(Sf"), there exists geH'^i^) with g"=hg if and only if 5*g" = 0. First suppose g" =hg,ge H^(M, Sf). Then there exists a c^ G Z ^ ( U , 5^) with [c''] = g. If we put /ic^ = c"^ g'^ = hg =-[c'"']. Hence by (3.83) we have 6*g" = [6c^] = 0. Conversely, suppose 6*g" = 0. Let g" = [c"^], and hc'^ = c"^ with C^G C^(U,^) and C ' " ^ G Z ^ ( U , ^ " ) . Then [6c^] = 6*g" = 0 in //^""'(M,^'), in other words, for a suitable SS
= [U^c'"] = [c'"'] = g".

Therefore (3.80) is exact. Proof of Theorem 3.7. We may assume that 5^' -4 5^ is the inclusion. Then H%5^') = r ( M , ^') is a submodule of / / ^ ( ^ ) = T(M, 5^). Hence the exactness of

132

3. Differential Forms, Vector Bundles, Sheaves

follows immediately from the exactness of (3.79). Moreover since (3.80) is exact, in order to prove the exactness of (3.78), it suffices to verify the exactness of W-\^") - ^ W{&") - ^ H^(^),

(3.84)

H'?(5^') - ^ //^(5^) — ^ W{&"')

(3.85)

and

for each g = 1, 2, Take an arbitrary g'e H ^ ( ^ ' ) . If g'=^''g" with g"G H ^ - ' ( ^ " ) , then by the definition (3.83) of 5*, g' = [Sc^"^] G H\S") for some c^"^ G C ^ - ^ U , ^ ) . Since t is the inclusion 5^''=-> 5^, t-g' = [6c^~^] is the cohomology class of dc"^'^ regarded as an element of Z^(U, 5^). Hence ig' = 0. Conversely, suppose ig' = 0. Let g' = [c'^] with c'^ G Z ^ ( U , 5^'). Then since [c'^] = tg' = 0 means that [c'^] = 0 as an element of //^(M, 5), there are a refinement 33 < U and an element c^"^ G C ^ - ^ S S , ^) such that ngc'^ = 8c''~\ Putting c"^-i = /jc^-\ we obtain Sc'""-' = hSc''-' = ng/zc'^ = 0. Hence c"^ e Z^-\U,^"). Therefore, putting g" = [c'"^"']G//^-'(5^"), we obtain from (3.83) 5 V = [5c^-^] = [ngc'^] = [c'^] = g'. Thus (3.84) is exact. Now we proceed to show that (3.85) is exact. Take any gG//^(5^). If g = Lg' with g' G //^(5^'), clearly hg = 0. Let g = [c^] with c^ G Z^( U, 5^). If [/ic^] = % = 0, there are a refinement 93 < U and an element C"^~^G C^~'(9S,5^") such that ng/zc^ = 5c"^"\ By Lemma 3.3, taking a suitable refinement SB < SS we can find an element c^~' G C ^ " ^ ( 2 B , 5^) for c"^~^ such that /zc^"' = n | c ' " ^ - \ Hence we have

that is, hiU^c^-8c''~') = 0. Therefore n^c^-5c^"'GZ^(S[B, 5^'). Consequently putting g' = [U^c^ - 8c^~^]e H'^iS^'), we obtain tg' = [U^c' - 8c'-'] = [U^c'] = [c'] = g since [8c''-'] = 0 in H ^ ( ^ ) . Thus (3.85) is exact.

I

For given i^-modules H^, H2,..., H ^ , . . . , Lj, L 2 , . . . , L^,... and Khomomorphisms K: H^^ H^+i, v\ L^-»L^+i, cp^: H^-^ L^, the diagram Hi-

^

h

-^ J%-^'

1^1-

'/'.

^2

^i V

• - ^ J^ v -

^1

>' -> J^ 2 - ^ -

> f

• - > I 'V

< ^ 1

k ~

>L,/ + 1

"•

(3.86)

133

§3.3. Sheaves and Cohomology

is called commutative if ^ = 1,2,3,.... If (3.86) is commutative, and each row is exact, we call (3.86) an exact commutative diagram. Similarly we define an exact commutative diagram of sheaves and sheaf homomorphisms. For example, ^

^

^

•

•A"

-^ ^ is an exact commutative diagram if each row is exact, L ° il/'= ij/<^ L and koil/ = il/"oh,

Theorem 3.8. An exact commutative diagram of sheaves 0->5^'

0->^

^

T'-^O

induces an exact commutative diagram of cohomology groups 0

> H\^')

- ^ H\&')

> H\9"')

r 0

> H\T)

> H\Sf')

- ^ •

r — U H^S-)

> H\T')

^ H\r)

- ^ •• • (3.87)

•^ H^'-'i^)

> W-\&'")

> H'^iT)

> H^(^)

>

Proof It suffices to verify the commutativity of (3.87). Since it is clear that k°il/ = if/"oh and Loil/'= il/°L, we have only to show that the diagram

134

3. Differential Forms, Vector Bundles, Sheaves

is commutative for q = 0,l, For any g"e H^{Sf"), by Lemma 3.4 there are C " ^ G Z ^ ( U , 5^") and C ^ G C ^ ( H , 5 ^ ) with g"-[c"^], /ic^ = c"^ such that 5*g" = [5c^] where 5c^ e Z^^'(U, 5^')- Hence (A'5*g" = [i^'Sc^]. Since t is the inclusion, L°II/' = if/ot means that ifj = \p' on ^' <^ ^. Therefore (/^'5c^ = (//5c^, hence (A'6*g" = [iA5c^]. On the other hand, we have iA"g" = [^"c"^] with ( A V ' ^ G Z ^ ( U , ^ " ) . Since il/"oh = koil/, we also have ifi'd"" ^i^'hc"" = kifjc\ Hence by (3.83), we have 5*iA"g'' = [5(Ac^], which implies that 5*(A"g" = fA'5*g". Since g" is an arbitrary element of H\&"'), we obtain 5*o ^" = ,/^'o 5* as desired. I

§3.4. de Rham's Theorem and Dolbeault's Theorem (a) Fine Sheaves Let /i: 5^-> 5^" be a homomorphism of sheaves over M. The closure of the set of points pe M with h(Sfp)9^ 0 is called the support of h and is denoted by supp h: supp/z=[{;7GM|;i(^^)7^0}]. Definition 3.13. Let 5^ be a sheaf over M M is called a fine sheaf if for any locally finite open covering U = { Uj}, there is a family of homomorphisms {hj}: hf. 5^^ 5^, such that (i) (ii)

supp hj c: Uj, Y.J hj = identity, i.e. Yj hj(s) = s for any

se^.

Note that by the condition (i), hj(s)9^0 for se^'p implies pe Uj. Consequently the summation in (ii) makes sense since U is locally finite. Example 3.2. ^ is a finite sheaf. Proof. Let U = {U,} be a locally finite open covering of M, and {p^} a partition of unity subordinate to U. For s =fp G sip, with / a local C°° function at p, define hjS by hjs = {pjf)p,

s=fp.

hjS is independent of the choice of/ In order to see that hj is a homomorphism, it suffices to verify that hj is continuous. Giv6n a neighbourhood U(hjS; g, L^) of hjS, since gp = / i / = (pjf)p, there is a neighbourhood V<^ U such that g = Pjd on V. Hence for te%L{s\f V), g = tEr(r) G V, ^7"^ = hjfq = (Pjf)q =Sq^ ^{hjS\ g, U).

§3.4. de Rham's Theorem and Dolbeault's Theorem

135

Thus hj is continuous, hence a homomorphism. It is clear that supp hj = supp Pj c Uj. Moreover X^. hj = identity follows from Xj Pj — ^- I- The essential point of the above argument is in the fact that i f / i s a local C°° function, so is Pjf. Consequently by a similar argument, we see that s^'', sd^''^ and s^{F) are fine sheaves where F is a C°° vector bundle over M. 0 is, however, not fine. In fact, if otherwise, for {hj} in Definition 3.13, hjleT{M, €) is holomorphic, where 1 is a holomorphic function which is identically one on M. Since hj\ is equal to 0 on M - supp hj, by the analytic continuation, /lyl = 0 on all of M, which contradicts Y.j hj\ = \. Theorem 3.9. If S is a fine sheaf over M, //^(M,5^) = 0 for q^\. Proof Since //^(M, 5^) = Uu.® n " / / ^ ( t l , ^ ) by (3.69), it suffices to prove that //^(U, ^ ) = Z^(U, 5^)/5C^-'(U, 5^) = 0,

^ ^ 1,

for any locally finite open covering U. We give the proof for the case q = 2, but the generalization is straightforward. Since 5^ is fine, there is a family {hj}, hj'. 5^^5^ of homomorphisms satisfying the above (i) and (ii). Each hi induces a homomorphism hi'. T{ W, 5^) -» r ( W, 5^) for any open set W^M. Let c^ = {o-jki} e Z^(U, 5^) be a 2-cocycle. For l/jnU^nUi?^ 0 , /ijO-yk: j? -^ {^i(^ijk){p) is a section of 5 over l/jnUj^n Ui, and if p^ supp hi, then {hiO-ij}^){p) = 0. Since supp /ij is closed in I7„ putting peUjnVkn

Ui,

rM-\^,^''''^^'^' pe UjnUk-supp

/i„

we get a section o-ij^: p-^ d-ij^{p) of 5 on L^n U^. &ijk is an extension of /ijO-yfc. For a fixed peUjn U^, orijj^{p)y^O for only finite Vs. Consequently putting i

we obtain T,^ G r ( U; n L4, '^) with r^y = -TJ^, that is, c^ = {TJ^} e C^{U, 5^). It suffices then to check c^=8c^. Since 8c^ = 0, we have o-jki = o-iu - o-iji + cT-yfc on t/j n Uj n [ 4 n (7/ T^ 0 . Hence ^i^jTc/ = hiCTiki - hidiji + /ijCTyfc.

136

3. Differential Forms, Vector Bundles, Sheaves

Figure 3

Since (7,^/ = ^/CTifcz, ^y/ = hia^i, (jyfe = hia^k on Uj nUknUin

^iki = ^iji = ^ijk = 0 on UjnU^nUiO'jkl -

Ui, and hiO-jki =

Ut, we have on UjnUkn TfcZ -

Uj,

Tji + Tjj^^

hence c^= 5c\ Now consider the sheaf si'' over a C°" manifold M For a germ s = cppO^ a local C°° r-form (p at;?, we define its exterior differential by ds = (d(p)p,

s = (pp.

d'.s^ds gives a homomorphism si''-^ si''^^ of sheaves, where we put si^ = si. The sheaf of locally constant functions C is a subsheaf of si^: C ^ si^. Then the sequence n d

,

d

0->C-

d

(3.88)

is exact where m = dimM, and i is the inclusion. Proof, For s=fpesi^, ds = (df)p = 0 implies that df=0 on some neighbourhood of /?, i.e., / is a constant there. Thus 0->C-> j ^ ^ -

§3.4. de Rham's Theorem and Dolbeault's Theorem

137

is exact. The exactness of sd'-^-^ M'-^ M''^\

r^l,

(3.89)

follows from Poincare's lemma (Theorem 3.2). In fact, take any s = (ppE sd\ where (p is a local C°° r-form at p. ds = (d(p)p = 0 implies that d(p = 0 on some neighbourhood U of p. We may assume that L/ is a coordinate multi-interval U = {Xp\\xl\ < e , . . . , \x^\ < s} with centre p. Then by Poincare's lemma there is a C°° ( r - l)-form ip on U such that (p = dif/. Putting t = il/pe sd'"^, we obtain s = (p = {difj)p = dt. Thus if ds = 0, there is a r G ^''"^ with dt = s. Conversely, if s = dt with tesd''~\ ds = d{dil/)p = {ddil/)p = 0. Consequently (3.89) is exact. I Definition 3.14. An exact sequence of sheaves over M O^^-^^^-i^'-^^^-i- ••

(3.90)

is called a fine resolution of 5^ if each S^' is fine. Thus Example 3.2 says that (3.88) is a fine resolution of C. Let d: SS -^ ^ " be a homomorphism of sheaves over M. Put 5^ = ker d. &^ is a subsheaf of %. We denote the inclusion 5^-^ ^ by i. Then

is exact. From this sequence we obtain Lemma 3.5. If ^ is fine, &") = H\M,

H\M,

dm)/dH\M,

W{M, 5^) = H'^-'iM, dm),

^),

g = 2,3,....

Proofi By Theorem 3.7, we get the exact sequence of cohomology groups 0-> H%^)•^H\m)-^

H\dm)

^

H\Se)^H\m)^"'

-^ /f^-i(^) ^ w\dm) ^ H^^) -^ H\m) -t.... Since ^ is fine, H^{m) = 0 for (^f ^ 1 by Theorem 3.9. Therefore we obtain the exact sequences 0 -> H\^)

-^ H\m)

0 -> H'-\dm)

-^ H%dm) ^ H\^)

^ H'(9') ^ 0 ,

-> 0,

q = 2,3,....

(3.91) (3.92)

138

3. Differential Forms, Vector Bundles, Sheaves

Hence

Theorem 3.10. / /

is a fine resolution of 9^ over M, then H'(M, 5^) = r ( M , dm'-')/dT(M,

SS^"'),

q^l.

(3.93)

Proof. From the exact sequence

we obtain the exact sequences

and r = l,2,3,....

0->d^'"'-^^'-^^^'->0,

By hypothesis each ^ ' is fine. Hence by Lemma 3.5, we have

H\dm'-')

= H\dm')/

H^dm'--')

= H''-\dm'),

dH\dm'-'), 9 = 2, 3 , 4 , . . . .

Therefore we have H ^ ( ^ ) = H^-\dm'')

= H'^-^dm^)

= H^d^^"') =

="

H\dm''-^)/dH\m''-'),

while by (3.70) we have / / ^ d ^ ^ - ' ) = r ( M , J ^ ^ " ' ) and

r(M,^^-0. I

f/^(^^-'):

§3.4. de Rham's Theorem and Dolbeault's Theorem

139

(b) de Rham's Theorem As stated earUer, the exact sequence (3.88) 6

_ d

.

d

d

gives a fine resolution of C over a C°° manifold M of dimension m. Consequently by Theorem 3.10, we obtain the following theorem. Theorem 3.11 (de Rham). H%M, C) = r ( M , dM^-')/dT{M,

^ ^ " ' ) , q^l.

I

(3.94)

Note that f/^(M, C) is the cohomology group of M with complex coefficients, depending only on the topology of M. As is well known, //^(M, C) = 0 for ^ ^ m + 1 . This is also induced from (3.88) easily. In fact, from the proof of Theorem 3.10, H^C) = W'^^idM"^-^), where dM"^'^ = d"^ is a fine sheaf. Hence by Theorem 3.9, //^(M, C) = H^-'"{s^'") = 0 for q^ m + l. Now we shall explain the meaning of the right-hand side of (3.94). Since for ^ ^ 1,0^ dsd^~^ -^sd^-^ si^^^ is exact, J^^~^ is the sheaf of local closed C°° g-forms. In fact ds^^~^ = keT d consists of germs peA^ with (d(p)p = d{(Pp) = 0, which means that d(p = 0 on some neighbourhood of/?. Thus r ( M , dsi"^'^) is the linear space of all ^/-closed C°° ^-forms on M, while J r ( M , ^^~^) is the linear subspace of r ( M , J^^~^) consisting of all exact r-forms on M. We call the quotient space H2(M) = r ( M , c?^^-')/^r(M,j2/^"'),

q^\,

the d'Cohomology group of M. Then de Rham's theorem:

Hl{M)^H\MX).

^^1,

implies that the cf-cohomology group /f ^(M), which is defined with respect to the differentiable structure ofM, actually depends only on the topology ofM. (c) Dolbeault's Theorem Let M = M" be an n-dimensional complex manifold, and ^^'^ the sheaf of germs of C°° {p, ^)-forms over M. The sheaf € of germs of holomorphic functions on M is a subsheaf of si = si^'^. By Theorem 1.6, a local C°° function / is holomorphic if and only if df=0. Thus 0-^(^-::>^o,o_4^o,i

(3 95^

140

3. Differential Forms, Vector Bundles, Sheaves

is exact, where t: €->si^'^ is the inclusion. Similarly O^ is a subsheaf of ^^'^, and a local C°° (p, 0)-form ^ is a holomorphic /7-form if and only if Sep = 0. Thus 0-^a^-^^^'^-i^^''

(3.96)

is also exact. ByDolbeault'slemma,a C°°(;?, ^)-formcp.q^l,onapolydiskis^-closed if and only if there is a C°° (p, ^ - 1 )-form ifj such that (p = ^ifj. Consequently

is exact. Thus by (3.96) and (3.97) we obtain the exact sequence 0 ^ a^ ^ ^AO ^ ^p,i _X

% ^p,n _^ 0.

(3.98)

Since ^^'^ is fine, (3.98) gives a fine resolution of Cl^ over M. For /? = 0, we obtain a fine resolution of €:

Theorem 3.12 (Dolbeault's Theorem).

H^(M, an=r(M, a^^'^-')/ar(M, s^^'^-'),

q^i,

(3.99)

Proo/ Apply Theorem 3.10 to (3.98). I In (3.99) a^^'^"^ is the sheaf of germs of a-closed C°° (p, g)-forms on M, r ( M , a^^'^~0 is the linear space of all a-closed C°° (p, g)-forms on M, and ar(M, si^'"^'^) its subspace. The quotient space

Hf''(M) = r(M, aj^^'^-')/ar(M, ^^'^-') is called the d-cohomology group of M. Thus Dolbeault's theorem says that //^(M, nn = Hf\M).

(3.100)

In particular for p = 0, (3.100) reads H\M,€)^HI'\M),

(3.101)

fl^ is the sheaf of germs of holomorphic sections of A^ T'^iM) over M: a^ = ^f A T'^iMU.

§3.4. de Rham's Theorem and Dolbeault's Theorem

141

Dolbeault's theorem can be extended for the sheaf 0(F) of germs of holomorphic sections of F over M, where F is a holomorphic vector bundle over M. Each term ^^'^ of (3.98) is the sheaf of germs of C°° sections of A'' T'^iM) A^ f*(M),whichwemayidentify with A'' r * ( M ) ® A^ f'iM). Thus

siP'^=Jf^ T*(M)®A t*(M)Y Similarly we define sd^'^{F) by ^^'^(F)=J^(F®A

7'*(M)y

where ^^'^(F) = ^ ( F ) . We shall show that €(F) has a fine resolution 0 -> ^ ( F ) -^ ^^'^(F) ^ j ^ ' ' ' ( F ) -^

^ ^^'"(F) -^ 0.

(3.102)

First we investigate the structure of sd^'^(F) and define d. Let {Cji,..., e^v} be a basis of F over Uj where U = {Uj} is a suitable locally finite open covering of M. We denote a point of M by z. Then each ejv(z)} e,;^: z ^ ^jxiz) is a holomorphic section of F over tT^, and {eji(z),..., form a basis of F^ for ZG UJ. LQtfjkiz) = (/jfcpt(^))A,/^=i,...,v be the transition function of F Then on UjnU^^^ (i>, ekAz)= i f%^z)ejAz),

(3.103)

A= l

where fjk^iz) is holomorphic. Assume that each Uj is a coordinate neighbourhood and let ( z j , . . . , zj) be the local complex coordinates on Uj. Then pxiq) sections q times

ej;,®dz^ A' ' • A dz],

A = l,...,^,

l^a<'-'
form a basis of F ® A^ r * ( M ) over U,. Consequently a C°° section <;p of F ® A^ T*(M) over an open subset W^ M is given on each Ujn W9^ (f> by ^ = i - ,

i


A = l ^ : a,...,7=1

where

142

3. Differential Forms, Vector Bundles, Sheaves

we can write (p as

A= l

where cpf is a C°° (0, ^)-form on Wn Uj. On

A=l

WnUjnUky^0,

K=l

hence by (3.103) we have .A -.=^ 1 f-f%^^)
(3.104)

If we denote Z l = i ^j\®^j t>y the vector {(pj,..., (pj), whose components are C°° (0, ^)-forms, (p is represented on each WnUj by the vector

which satisfies (3.104) on WnUj-n U^. Conversely, suppose given on each WnUj7^0 a vector (cp],..., (pJ) with cpj a C°° (0, g)-form, satisfying (3.104) on Wn UjnUj,, Then there is a C°° section cp of F ® A ^ f * ( M ) over W such that <^ = Z l = i ^j\®^j on Wn Uj 7^ 0 . (3.104) gives the same transition relation as that of F: Cj — L

fjk/jii^j^k'

Accordingly we call a C°° section (p = ( ( p ] , . . . , (pJ) of F ® A^ f *(M) a C°° (O,q)-form with coefficients in F, and ^^'^(F) r/ie sheaf of germs of C^ (0, q)-forms with coefficients in F. For ^ = 0, each (p^^ is a C°° function. For a C°° (0, g)-form cp = {(pj,..., (pJ) with coefficients in F, we define ~dcp:=(d
(3.105)

Since ^/^^^(z) = 0, applying d to (3.104) we obtain ~dcpf= i

fjk^izfdcp^,.

Therefore dcp is a C°° (0, ^ + l)-form with coefficients in F. Thus d gives a homomorphism

a:^^'^(F)^^^'^^H^).

§3.4. de Rham's Theorem and Dolbeault's Theorem

143

Since s^^'^(F) is fine, in order to show that (3.102) is a fine resolution, it suffices to prove the exactness of (3.102). For this it suffices to prove that at each point zeM, the sequence of the stalks 0 -> €(F), -^ si'^'^F), -^ ^""''(F), - i • • • is exact. Thus we have only to prove the exactness of (3.102) on each Uj. But on Uj, a local C°° (0, ^)-form (p = ( ^ ] , . . . , (pj) with coefficients in F may be regarded as a z^-tuple of C°° (0, ^)-forms (Pj, A = 1 , . . . , K Moreover d(p = (d(Pj,... ,d(pj). Therefore the exactness of (3.102) follows from the exactness of o-^(D'-^^^'^-i j2/^''^ — ^ j2^^'"->o.

Applying Theorem 3.10 to (3.102) we obtain immediately the following theorem. Theorem 3.13. ff^(M, 0(F)) = r ( M , a^^'^-^(F))/ar(M, ^ ^ ' ^ - ^ ( F ) ) ,

q^l.

I

(3.106) Let (p be a C°° section of F ® A'' T'^(M) A^ f *(M) over an open set Wc: M. Then

f*(M)y

If each (pf of (p = {(pj,... ,
T'^iM)).

Substituting F ® A^ T'^iM) for F in (3.106), we obtain

//^(M, a^(F)) ^ r(M, aj2/^'^-'(F))/ar(M, j2i^^'^-'(F)),

g ^ i,

(3.107)

where r ( M , a^^'^(F)) is the linear space of all a-closed C°° (p, ^)-forms

144

3. Differential Forms, Vector Bundles, Sheaves

with coefficients in F over M. The quotient space Hf\M,

F) = r ( M , dsd''^-\F))/dr{M,

^^'^"H^))

is called the d-cohomology group of M with coefficients in F. With this notation, (3.107) is written as H^(M, n^(F)) = Hl'\M,

F),

(3.108)

§3.5. Harmonic Differential Forms (a) Inner Product, Norm, Dual Form Let M be an n-dimensional complex manifold, { z j , . . . , z^,...}, Zj\ p -» Zj(p) = ( z ) , . . . , z") local complex coordinates on M, and Uj the domain of Zj, and suppose that U = {Uj} is locally finite. Suppose that p^z{p) = ( z \ . . . , z") is an arbitrary local complex coordinates system on M. We represent a point of M by its local coordinates z = z{p). In the sequel, we represent a tensor field by its components with respect to local coordinates ( z \ . . . , z""). Thus, for example, by a "tensor field o-^^", we mean a section Z

(Tly—®dz^®dz'^

of T{M)®T''{M)®f''{M). If we denote by crf^y the component of a tensor field aly with respect to the local coordinates ( z ] , . . . , z"), we obtain

Definition 3.15. A C°° covariant tensor field YJL,^=\Sa^ dz'^®dz^, Sap = gapiz), is called a Hermitian metric if g^a(z) = ga^iz) and the Hermitian form Za,)3=i Scx^(z)^'^[^ with complex variables ^ \ . . . , ^", is positive definite at every point of M Z ^a^ dz"'®dz^ is called a metric because by virtue of it we can define the length VZ gapr[^ of a tangent vectorJ = Z« ^"(a/az")G r , ( M ) . We associate to the Hermitian metric Z«)3=i ^a^ dz"'®dz^ a (1, l)-form ^ =^ Z

gc.pdz'" Adz^,

i=V-l.

§3.5. Harmonic Differential Forms

145

Since (o = -i I g ^ ^z" A dz^ = i X gfia dz^ A dz"^ = (o, w is a real (1, l)-form. Theorem 3.14. Given any complex manifold M, we can introduce a Hermitian metric Xa,^==i gap dz"'®dz^ on M. Proof. Let U = {L^} be a locally finite open covering of M, and {pj} a. partition of unity subordinate to it. py = py(z)^0 is a C°° function on M with supp Pj c: Uj, and X^ Pj{z) = 1. Put n

^j(^) = i Z Pj(z) dzj A dzj. 7= 1

Then w^ = (Oj(z) is a C°°real (1, l)-form on Uj, and (Oj{z) = 0 on Uj-supp Pj. Consequently putting a)j(z) = (Oj(z) for ze Uj, and a5;(z) = 0 for z^ Uj, we can extend coj to a C°° (1, l)-form aij = c!)j(z) on M. Put

Since supp cOj = supp py c: Uj and {[7,} is locally finite, this summation makes sense. Therefore w is a real C°° (1, l)-form on M with (o = i I

g^pdz'^Adz^,

gp- = g^^,

We claim that Z2,^=i ga^ ^^" ® ^^^ is a Hermitian metric. For this it suffices to see that Za,/3=i ga/3(z)^"f'^ is positive definite at each zeM. By the definition of (o, i

g^^(z) dz- Adz^=

a,(3 = l

I UjBz

i

pj(z) dzj A dzl

(3.109)

7=1

where the summation Y^UBZ is taken over all j with zeUj. Consider (^\ . . . , ^'') as the components of a tangent vector ^e T^iM) with respect to (z\ . . . , z"), and denote by ( ^ ] , . . . , (") the components of ^ with respect to (z],..., zj). Then since the transition relation ^7 = Xa (dzj/dz'^)^" of the components of a tangent vector is the same as that of the bases of T'^iM): dzJ = Y^^ {dzj/dz'^) dz"", we obtain from (3.109) immediately

i

g.^{z)CC'= I i PjizMj?'

oi,P = l

UjBz

7=1

Therefore I"^^=i ga^(z) dz°'®dz^ is positive definite.

I

146

3. Differential Forms, Vector Bundles, Sheaves

Let M be an n-dimensional complex manifold endowed with a Hermitian metric YJl,p^\ Sapdz"" ® dz^. Let n

^ =i I

8a^ dz"" A dz^

a,l3 = l

be its associated real (1, l)-form. Put g{z) = det (g«^(^))«i8=i n fold

«• We denote

(o A ' ' ' A (o by co". Then by an elementary calculation we obtain o)" = (iynlgiz)

dz'Adz'A"'A

dz"" A dz"".

(3.110)

For local real coordinates {x\ . . . , x^") with z" = x^"~^ + /x^", we have / Jz" A J r = 2 ^x^"-' A dx^",

a = 1 , . . . , n.

Hence we have — = 2^giz)dx'A"'Adx^\ n\

(3.111)

Since X ga^(z)("[^ is positive definite, g ( z ) > 0 . Therefore using co^'/nl as the volume element, we define t/ie integral of a continuous function f(z) on Mhy f{z)2"g(z)

dx' • " dx^\

(3.112)

where g(z) is the component gi2---m(^) of the skew-symmetric covariant tensor field of rank 2n with m — 2n. We denote by (g"^(z))«,^=i .,„ the inverse matrix of (g«/3(^))a=i,...,nI g'^{z)g,,{z)

= 6;,

I g.0(z)g^^{z)

= 81.

g"^ = g«/^(2) gives a C°° contravariant tensor field on M. In fact with respect to the coordinate change z^ -> z^, the transition relations of the component gja0(z) of gap(z) are given by gj.^{z)=

Z T-^ T-7^/^A^(^)' ,}=,dzj\dzf)

and (a^7/^^fc)A,a=i,...,n = (dzi/dz]')^lp^,_ „. Consequently, (gfiz)) = (gj^0{z))-' and ( g ^ ( z ) ) = ( g , , ^ ( z ) ) - \ we obtain

sr{^)= I fe &^."'(-)-

(^-ll^)

putting

(3.114)

§3.5. Harmonic Differential Forms

147

Let, for example, cp =IZ",^,y=i Zj, the transition relations of (p^^py are given by

From this and (3.114), it is clear that for C°° (2, l)-forms
is a C°° function of z, independent of the choice of local coordinates. If we put

then we have 1

"

Similarly for C^ (p, q)-forms (p =——T cp^^... p^...0 (z) dz""^ A ' • ' A dz'^p A dz^^ A ' • -A^z^s plql ( A = - y - ^ I t A a r " « p ^ i - ^ - , ( ^ ) ^ ^ " ' ^ - • • A iz"p A C?Z^^ A • - A ^ A

we put (cp, .A)(z)=-^Z
(3.115)

where

TTzen ((p, «/^)(z) is a C^ function of z. We define the inner product of (p and iff by

(^,^)= I

(9,^)(^)^. n

(3.116)

148

3. Differential Forms, Vector Bundles, Sheaves

The inner product satisfies the following properties: (i) (ii)

iip,cp) = (cp,il;). {(p, (p) ^ 0. (
In fact for any ZQG M, choose local coordinates ( z \ . . . , z") such that ga^-(^o) = ^ap' Then g^'"(zo) = 5,^, and ^_ p\qV

{(P, < A ) ( Z o ) = - y - ^ I

(pa,...a^p,-pS^o)^c.,-ap^,-0^iZo),

From these equalities (i) and (ii) follow immediately. Also it is clear that {(p, iff) is linear in (p. We call J{(p,(p) the norm of

For each (p, q)-form if/, we can find an {n—p,n-

q)-form ^ip such that

{cp,ilj)(z)^=cp{z)A^ij;{z). n\

(3.117)

Proof. For simplicity we denote as follows. Ap = a,"'ap,

Bq = p,'"pq

with

dz^p = dz^^ A • • • A dz\

a,<"'
p,<"'
dz^^ = ^zf M • • • A dz'^^.

Moreover for A^ = ai • • • a^, we put

where a^+i < • • • < « „ , and { a i , . . . , a^, a ^ + i , . . . , a„} is a permutation of { 1 , . . . , n}. Similarly we define B^-q for a given Bq = p^- - - Pq. Then with this notation we can write

^^

/-ai

•••

' ^dz

^.

Ary,Ba

Let (Ap

A„_A

apttp+i

•••

OLA

§3.5. Harmonic Differential Forms

„

149

where sgn denotes the signature of the permutation. Then by (3.110) — = ( / ) « ( - l ) " ( " - l ) / 2 g ( ^ ) dz'A"

'Adz''

Adz'A"

'A

dz\

Since dz'

A • • • A dz"" A dz'

A • • •A Jz"

= sgnl

^

= sgnl

^

" I ^z"i A • • • A c/z"« A dz^^ A • • • A ^z^"

" ~ ^ I J z ^ P A 6?Z^"-P A Jz^-? A ^ Z ^ " - %

putting /c = 5 « ( n - l ) + ( n - l ) g , we obtain ^ = (f)«(-l)^ sgn(^^

^"~'')g(z) dz^p

A~d^^dz^^-pAdz^^-^.

On the other hand, by (3.115) (
I

^^^5-^(z)(A^p^
Hence ((^,(A)(^)—=(0"(-l)' [Ap

ASgn /

I

Cp^^BSz)dz^PAdz' An_p\

^"-Mg(z)(A^A(z)c^z^"-AJz^"-..

Put Mp = fjLi ' ' • iJLp with 1 ^ /ii < )L62 < • • '
For in this case some /x, coincides with one of a ^ + i , . . . , «„. Similarly for '
{cp,^){z)—={in-ir Ap,Bq

I .pM„;.,(z)c/z%Adz~. \Dq

= ^A(0"(-1)'' I

Dn-ql

sgnf^"

^"-'')g(z).A^A(z)dz^"-.AdP^.

3. Differential Forms, Vector Bundles, Sheaves

150

Hence defining *i/^ by sgn(^^

*(A==(0"(-1)' I Ap,Bq

\Dq

^"-^)g(z)eA^A(z)Jz^«-.A^P^,

(3.118)

rin-q/

with k = \n{n — \) + {n—p)q, we have (3.117).

I

*(/^ is independent of the choice of local coordinates used in (3.118), since so is (
J_

' ^ = - 7 — f I « A « i - « p ^ i - ^ , ( ^ ) ^ ^ " ' A - • • A dz'^P A dz^^ A • • • A JZ^'J

p\q\

is a (q,p)4orm.

If we write

then
l)^^
Hence putting

we have ^«i-«p^r-^~,(z) =

(-l)^^^^"i-^>i—p(z).

Thus (3.118) is rewritten as A„,B.

\ 5 f^ql

5^ nn - q

X g ( z ) ( / ^ V p ( z ) C/Z^"-P A ^^Z^"--^.

Replacing if/ by ijj and interchanging A with B, we obtain

*^ = (,r(-l)"^"-^'^^""'' Z sgnf^" ^"-') Ap,Bq

xg(z)ilA^^(z)

\tSq

dz^^-^ A dz~^,

Dn~qJ

(3.119)

where

We call this *i/^ the dual form of a (/?, ^)-form j/^. *(/^ is an {n — q,n —p)4oTm.

§3.5. Harmonic Differential Forms

151

Clearly the map if/^^if/ is linear, namely, we have *(Ci(p + C2eA) = Ci*(;p + C2*eA

fo^* Cj, C2GC.

We have ^=*iA.

(3.120)

Proof. It suffices to check this at any point ZQ G M . Take local coordinates (z\ . . . , z") with centre Jo such that gapi^o) = ^ap- Then g"^(zo) = 5«^, hence in (3.118) we obtain (/^"^^^^ZQ) = IAA^B,(^O)- Similarly in (3.119) we have ilf^p^^izo) = II/A B (ZQ)' Moreover we have

Substituting these equalities into (3.118) and (3.119), and comparing them, we obtain ^^^(iJ = (-l)«+"/>+(«-^)(-/'H(«-/')^*^(zo)-*(A(^o).

I

We denote a (/?, ^)-form if/ by (/^^^'^\ Then **e/,(^'^) = (-1)^^ V^'''^^.

(3.121)

Proo/ Again we have only to check this pointwise. Fix ZQGM and take local coordinates as in the proof of (3.120). Then since g(zo) = 1, we have by (3.119) (*^)B„_,A._,(zo) = (0"(-l)"^"-^^/^"-sgn(^^

^yyA^sS^o),

hence,

__ /_-j^\n + np+n(n-q)+(«-q)(3r+(n-p)/? ,

- fz )

= (-1)^"VA,B,(^O).

Hence **i/f = ( - l ) ^ ' ' V .

I

By (3.117) the inner product ofcp and ij/ is given by ((p,il/)=

(fA^iii. JM

(3.122)

152

3. Differential Forms, Vector Bundles, Sheaves

(b) Harmonic Differential Forms Let M be a compact complex manifold of dimension n, and let i^^'^ = r ( M , ^^'^) be the linear space of C°° (/?, g)-forms on M,d\(p-^ dcp is a linear partial differential operator of order 1 which maps if^'^ into if^"^^'^, and d: (p ^d(p is 3. linear partial differential operator of order 1 which maps if^'^ into i^^'^^\ Define b = -*a*.

(3.123)

Since *: (/f-. *,/^ maps i?^'^ into i^""^'""^, b ma/?5 L^'^, ^ ^ 1 mro i^^'^-\ For i/f G i^^'°, b(/^ = 0. b is also a linear partial differential operator of order 1. For (p e <^^'^~' and (^ e i^^'^ we have (a^,(A) = ((p,b(A).

(3.124)

Proo/ Since M is compact, we have by Theorem 3.1,

[ d{

0.

J M

Since

A^ii/)=d(p A^ii/-(-iy^'^(p

A5*i/f.

On the other hand by (3.120) and (3.121) *bj/i^= -*(*a*^) = -**a*(/^ = (-1)^'^^5*«/^. Hence d{(p A^iff) = d(p A^ilf —q> A *bi/^.

Hence

JM

JM

Therefore by (3.122), (5
(3.125)

§3.5. Harmonic Differential Forms

153

The coefficients i/^ar cp^i ^^(^) of a C°° (p, g)-form if/ form a C°° covariant tensor field, while

define a contravariant tensor field. Accordingly we call
(b(A)^"^^-^
(3.126)

^ = ig(z) az'^ Proof. Let ( z \ . . . , z") be local coordinates on a coordinate neighbourhood U. It suffices to prove (3.126) for U. Put the right-hand side of (3.126) as ^^^2 •^'?(z), and let ^ be the (;?, g - l ) - f o r m on U whose contravariant components are ^^^^ ^-^Cz). In order to prove that b(/^ = ^, it suffices to show that (b(/^, (p) = (^, (p) for any C°° (/?, g - l)-form (;p with supp (p^ U. For this, since {hif/, cp) = (ij/, d(p) by (3.125) it suffices to verify

Put (PAPi---Pq(z) =
Hence

q\ A,I3„...,I3^

If •

5= 1

1

= 7—77^

I

I V'^"^^-^'(z)5,^^-,,..,,(z).

Put B = y32--)S„ l S j 8 2 < - - < / 3 , S n . Then

A,B p = \

154

3. Differential Forms, Vector Bundles, Sheaves

Therefore by (3.112) (^, ~dcp)=\

I

i

r^\z)d,
dx'"'

dx^\

J M A,B /3 = 1

Since supp (p'=^ U, and U is a. polydisk, by integrating by part, the above integral is equal to

[ I I a^(g(z)^^^«(z)).p^-H(z)2"g(z)

dx'---

dx'

J U A,B p = l

I E

-T\^,(.g{^)'}>'^\x))^^s{z)—.

Hence

{ii^rdcp)=\ I t^(z)
nl

(^,
nl

Thus{il;,d(p) = i^,
(3.127)

n is called the complex Laplace-Beltram operator. D: ^ - ^ D ^ is a linear partial differential operator of rank 2. We denote the covariant components the covariant components ^Ai • Api?i •i?g(^) of (p by (PAN(Z), {Hep) ^^...^^0^...0^iz) of Dip by {n
-z:.^=x g^-izXdVdz" dz^). More precisely we have

(n.p)^fi(z) = - X g^'-dj^cp^siz) a,/3 = l

+ I ( E a ^ f X + i a\§% + bX§)
\a = l

13 = 1

I

§3.5.

Harmonic Differential Forms

where aXs", tives.

^AS^^,

155

b^s are polynomials ofg^^, g^" and their partial deriva-

Proof. Since D = -^*(9* — *5*5 and * is an operator which makes a linear combination with coefficients in polynomials of g«^ and g^", in order to determine the principal part of D, it suffices to determine the terms of (n(p)As(^) which contain no partial derivatives of g«^, g^". For this purpose, we may assume that ga^ = ga^(z) are constants. If Sap are assumed to be constants, g^" are also constants. Hence by (3.126) (b(p)^^^-^
dpcp^^^^-Mz).

Therefore (t>(p)A^r-P,(^) = - l dp(p^0^...p^{z)

7 /3

Hence

(5b^)A^,..^-,(z)=I g^^ i

{-\Y~d,d,cpA,,^...^^...^^{z).

On the other hand

(ba(p)A^,..^-,(z)=(-ir^ I g'%(5cp)A.^,..^-,(^) y,7

(3,7

s= l

Since d^^df^ =dpdp^, we have n

Thus

(•
g^"'dadf3(PAB{^) if ga^ ^rc constants. Therefore the

principal part of D i s ' - I ^ ^ ^ j g^"(z)((9V5z"5z^).

I

156

3. Differential Forms, Vector Bundles, Sheaves

Corollary. The partial differential operator D is strongly elliptic. (For definition, see Definition 4.2 of the Appendix). I n is formally self-adjoint, that is, (n(p,(A) = (^,niA)

(3.129)

holds for any cp, if/e^P'^. Proof By (3.124) and (3.125) (D (p, (A) = (dt>(p, ip) + (hdcp, ijj) = (b(p, b(A) + (dcp, dijj) = i
(3.130)

Here for (pe^^'", we have (n(;p, (p)= ||b(^||^ since d(p = 0. Similarly for (p= 0. If (p is harmonic, n(p = dh(p-\-hd(p = 0. Conversely, by (3.130) if[I\(p=0, (p is harmonic. Thus (p is harmonic if and only if[ll(p = 0. We denote by H^'^ the linear subspace of i^^'^ consisting of all harmonic (/?, g)-forms. W'^ = {(peLP'''\n(p = 0}. Since M is compact and D is strongly elliptic, H^'^ is finite dimensional (See Theorem 7.3 of Appendix.) We say that (p and iff are orthogonal if {(p, lA) = 0. If (JP e H^'^, ((^, De^) = {U(p,ilf) = 0 for any i/^ei^^'^ hence, H^''^ and Dif^'^ are orthogonal. We denote the direct sum of mutually orthogonal subspaces by @. The following theorem is fundamental. Theorem 3.17. if^'^ = H^'^enif^'^.

I

(3.131)

For proof see Appendix (Corollary to Theorem 7.4 of Appendix). Corollary. ^ A ^ ^ H P , 9 0 5 ^ A 9 - i + ^^A^+i^

where we excise the term 5i^^'^~^ for q = 0 and t>^^'^^^ for q = n.

(3.132)

§3.5. Harmonic Differential Forms

157

Proof. It is clear from (3.124) and (3.125) that H^'^ d^^^^'' and bi^^'^"^' are mutually orthogonal. For any (peH^'"^, by (3.131) there is an / I G H ^ ' ^ and i/fGi^^'^ such that

Since Oip = d\)ilJ-^tdil/, hil/e^P'^~' and ^JAFG i^^'^'^\ we obtain (3.132).

I

Consider the a-cohomology //f^(M) = r ( M , aj2F'^-0/ar(M, ^ ^ ' ^ - ' ) , g ^ l , of M. Then r ( M , ^^'^"') = if'''^"\ and r ( M , a^^^'^"') is the linear subspace of if^'^ of all ^-closed C°°(/?, ^)-forms. From (3.132) we have

r(M, a^^'^"0 = H^'^ea^^'^"\ In fact for any cpe^^'^ represent cp as (p = h + dil/-\-hr] by (3.132) where h G H^'^, (A e if^'^~' and 77 G if^'^""'. Then if b77 = 0, ^cp = 0 clearly. Conversely, if 5

= W'^.

(3.133)

In other words a harmonic form (p e H^'^ gives a representative of a dcohomology class of (p, q)-forms. By Dolbeault's theorem (3.100), we have H^(M, a^) = Hl'\M),

q^l.

Hence we obtain the following Theorem 3.18 (Hodge-Dolbeault). H^M,aP) = W'\

I

(3.134)

For a C°° (;?, 0)-form (p, d

158

3. Differential Forms, Vector Bundles, Sheaves

such that 7r~\ Uj) = (7, x C" for each Uj, Let ( ^ ] , . . . , ^J) be the fibre coordinates of F over Uj, andyjfc(z) = (/^fc^(z));^^=i ,, the transition function of F. Then on L^ n t/^ T^ 0 , ^j=l

fjkAz)a-

(3.135)

Definition 3.18. A Hermitian metric on the fibres ofF is defined by specifying on each Uj a positive definite form

such that aj^Jz)

is C°° and that I

aj,^(z)cn^=

A,/u. = l

Z

a,,^izUU^

(3.136)

A,yu, = l

for ze UjC) U^. By (3.135), (3.136) is equivalent to ak.^{z)=

I

frkdz)ffk^iz)aj^^{z).

(3.137)

Therefore, in view of the transition relations (3.50) of F*, ajxfiiz) represents a C°° section of F * ® F * where F* is the conjugate of the dual bundle F*. Let i^^'^(F) be the linear space of all C^ip, ^)-forms with coefficients in F:i^^'^(F) = r ( M , ^ ^ ' ^ ( F ) ) . (pe^^^^F) is represented by a vector (p = {(pj,..., (pj) on Uj where (pj are C°°(jp, ^)-forms satisfying, on ^•nt/,^0,
(3.138)

For (p,il/e ^P'%F), Y,l^=, aj^fi{z)(pj A *(Aj' is a C°° 2n-form on Uj. Comparing (3.138) with (3.135), we see from (3.136) that, on UJH U^T^^, Z A,/x = l

aj^f,{z)ip^^^lP^=

Z

ak^^{z)
A,/x = l

holds. Consequently there is a 2n-form on M which coincides with Z aj;,^{z)(pj A^ij/^ on each Uj. We denote it by ZA,^ ^ A / I ( ^ ) ^ ^ A*I/?^ The

§3.5. Harmonic Differential Forms

159

inner product of (p, (/^G i^^'^(F) is defined by {
Z

a,^(z)(p^A*(^^

(3.139)

It is easy to verify that (i) {ip,(p) = i(p, i/f) and that (ii) (-\F), ~d

a''{z)*d(i

T=l

a^,iz)*i(,A.

(3.140) /

\ M = 1

ha is the adjoint of d, that is, (a^, (A) = (
^ e L^'^^-^F),

(A e L^'^^CF).

(3.141)

In fact, writing a;^^ for a^^(z), we have Z «A^^' A*(b.(A)^ = Z
A

\ fx

J

= (-l)''"'Z^'A5(la,^*r). Since Y.^,^ «AM
\A,At

/

\A,yu,

= Z

/

OAMS*'' A

* ^ + (-i)^-^''-' Y.
A,At

A

\ /Lt

Hence we have d( Z ci,^
a,^ A *(b,(A)'

Integrating over M, we obtain (^(p, i^) = (<^, bat^)ba is an analogy of b = —*5*. As in the case of b, we have (ba^A,
/

160

3. Differential Forms, Vector Bundles, Sheaves

By (3.140) we have (hAr-H"-

i a'\z)^l i aa^,(z)A*(/.^). T=l

(3.142)

/

\M=1

We define • a = 5t),+b,a

(3.143)

analogously to D. It is clear from (3.142) that the principal part of Da coincides with that of n , hence is equal to -Z«,^=.i g^"(z)(aV^a"^^^). Since (n,(p,^) = ( 9 , n . ^ ) ,

(3.144)

Da is a self-adjoint strongly elliptic linear partial differential operator. (p e i^^'^(F) is called a harmonic form with coefficients in F if 'dcp =t^a

(3.145)

I

(3.146)

Corollary. i?^'^(F) = H^'^(F)ea^^'^-^(F)eb«^^'^-"H^).

•

(3.147)

Consider the ^-cohomology with coefficients in F : Hl'\F)

= r ( M , a^^'^-HF))/ar(M, J2^^'^-^(F)).

r ( M , dsd^'^~\F)) is the linear space of all (p e if^'^(F) with d(p = 0. Hence by (3.147) r(M,a^^'^-^(F)) = H^'^(F)e5i^^'^-H^). Since T(M, ^^'^-'(F)) = if^'^-H^), we have /ff(M,F)-H^'^(F).

§3.5. Harmonic Differential Forms

161

Thus by (3.108) we obtain the following theorem. Theorem 3.20. H%M,nP(F))

= W'%F).

Corollary. //^(M, {1^(F)) is finite dimensional.

I

(3.148)

I

Let F * be the dual bundle of F, and (rjji,..., rjj^) the fibre coordinates of F * over Uj. Then the transition relations are, by (3.50), VK^=

i fjkAz)vj.,

(3.149)

A= l

where {fjkfxi^)) is the transition function ofF. For il/ = {il/],..., if/J) e i^^'^(F), put 4^%= i Then since on UjnU},9^0,

aj,^(z)^4i^.

ipl-= Y.l=Jfkr{z)Vk.

(3.150) we obtain from (3.137)

A= l

Comparing this with (3.149), we see that (i/^*i,..., IIJ%) is a C°° (n -;?, n g)-form lA* with coefficients in F*: i/f* = (j/^.^i,..., «A,*J e i^"-^'"-^(F*). Solving (3.150) with respect to i^jf, since ^^i/^j" = ( - l ) ^ ^ V j ' , we obtain

^r=(-i)^"^ I «^(^)*^fA. Thus the map (A-» j/^* maps i^^'^(F) onto i^"-^,'"-^(F*) bijectively. This map is U-linear. In terms of the components with respect to arbitrary fibre coordinates ( r / i , . . . , 77J, U\ . . . , D of F and F * w i t h ! , ^ S , = Z , ^^r^,,, we write (3.150) as

For (p e ^P'^(F) and i^ e i^"~^'"~^(F*), X L i ^A,* A (pj is a C"^ 2n-form on Uj, and on UjnUkT^ 0 , we have

A

A

which implies that XA ^t ^^^ is a C°° 2n-form on M.

162

3. Difierential Forms, Vector Bundles, Sheaves

Theorem 3.21. Define the inner product of4>* e H""^-""«(F*) and (p e by

{r,
W'^F)

1
M A= l

Then with respect to this inner product H" ^'" ^(F*) is the dual of H^'^(F). Proof. Define the inner product of (/f* e i^"-^'"-^(F*) and (pe^^^^F)

by

{^^
A

Then since il/t = Y.^ ^A^tC^)*^'^? we have

Introducing a Hermitian metric XA,/X ^^^i^)V\Vtx on the fibres of F*, we define the inner product of (A*, <^* e ^ " - / ' ' " - ^ ( F * ) by ^'{z)iPtA^Cpl. J M A,/u,

Since Z a'^^(z)*<^* = Z a'^^(z) I a.^(z)**^^ = ( - 1 ^ ^ ' , we have

1 I M

M A

{il,*,^*) = {tl,*,
i'P,n

Hence

For (p € ^"'"-'{F) form on M. Hence

and ./»* e iP"-''-"-«(F*), I^, .p^ A ./^J is a C°° (n, n - 1 ) -

J M \ A

/

J

M

\

/

A

Consequently integrating the equaHty

\ A

(3.151)

/

A

A

§3.5. Harmonic Differential Forms

163

over M, we obtain ((A*,ab> = (-ir^(a(A*,

.

(3.152)

Since b^ is the adjoint of d, we have, by (3.151)

On the other hand, (ai/^*, (p) = (dilj'^, cp"^) holds, hence by (3.152), (a(A*,
^ e ^P^'^iF).

(3.153)

Similarly since (5(/f*,
^ e ^^''^(F).

(3.154)

(3.153) and (3.154) imply that (/^* is harmonic if and only if \p is harmonic. Thus i//^ (A* maps H^'^(F) bijectively onto H"-^'""^(F*). Choose an orthonormal basis {^i,..., e^} of H^'^(F) where m = dimH^'^(F). Then { e f , . . . , e*} is an orthonormal basis of H"-^'"-^(F*). For, by (3.151) (ef, e^) = (efc, e,) = 5,fc, and for any (/^ = Xr=i ^fc^fc^M^'^(^)? we have «A* = Zr=i^^fc- Since by (3.151)

we have m

for
164

3. Differential Forms, Vector Bundles, Sheaves

Theorem 3.22. Let M be a compact complex manifold M of dimension n, and F a holomorphic vector bundle over M. Then the equality dim //^(M, n^(F)) = dim H"-^(M, a""^(F*))

(3.155)

holds where F* is the dual bundle of F. Proof Since by Theorem 3.21, dimH^'^(F) = dim H"-^'"-^(F*), (3.155) follows from (3.148) immediately. I Let 0 be the sheaf of germs of holomorphic vector fields: % = €{ T{M)), where T{M) is the tangent bundle of M. Since €{T{M)) = a^{T{M)), by (3.155) dim W{M, 0 ) = dim //""^(M, n " ( r * ( M ) ) . Moreover we have fl" = ^(A"F*(M)). Let {L^} be a finite open covering of M consisting of coordinate neighbourhoods Up and {z],..., z") the local complex coordinates on Uj. Then dz] ^' - - ^ dz" is a basis of A" ^ * ( ^ ) over Uj. A" F*(M) is a vector bundle of rank 1, and each fibre A" Ff (M) at z e L^ is a 1-dimensional linear space C dz] A- - • A dzj. Put

Jjkiz) = det—I

—

d{Zj,...,Zj)

Then since dzl A • • • A dzl = Jjk(z) dz] A • • • A dz], the transition relations for the fibre coordinates on A" T'^iM) are given by Cj = Uz)^j,.

(3.156)

A" T^(M) is called the canonical bundle of M, and is denoted by K: i^ = A" T'^iM). K plays an important role in the theory of complex manifolds and algebraic geometry. Since

a"(r*(M)) = ^(X0 r*(M)) = ^(F*(M)® x) = n\K), we obtain the following equality dim f/^(M, @) = dim //""'^(M, n\K)).

(3.157)

§3.6. Complex Line Bundles

185

§3.6. Complex Line Bundles (a) Complex Line Bundles By a complex line bundle we mean a holomorphic vector bundle of rank 1 over a complex manifold. Let (F, M, TT) be a complex line bundle. We choose a finite open covering U = {Uj} such that 7r~^(Uj)= UjXC Then (z, ij) e UjXC and (z, ^k) e U^xC are the same point on F if

where the transition function yjfc(z) is a non-vanishing holomorphic function on Ujn L/fc. Moreover we have fjj{z) = 1, fkj(z) =fjk(zy^ by (3.40) and on UinUjnUj,9^0,byi339), fiu{z)=fij{z)fju{z).

(3.158)

Let 6^ be the set of germs of non-vanishing holomorphic functions at peM. 0^ forms a group with respect to the multiplication. Since 0^ is an Abelian group, considering it as a Z-module, we can define a sheaf ^* = UpeM ^p of germs of non-vanishing holomorphic functions over M. ^* is an open subset of €. Let g = g(z) be a local holomorphic function oh M. We denote / = / ( z ) = e^"'^^"^ by e(g). Then ^ : g - > / = e ( g ) defines a homomorphism gp-> e(g)p of 0 into ^*, which we denote also by e. Since ^27rig(z)^^ if aiid only if g{z) is a Z-valued locally constant function, ker e = Z, hence, 0-^Z-^^-i^*-»0

(3.159)

is exact. By Theorem 3.7 we obtain from this exact sequence the following exact sequence of cohomologies. > H\M, 0) -> H\M, 0^) ^ H\M, Z) -> • • •.

(3.160)

If we regard the transition function fi^ =fjk{z) of a complex line bundle F = (F, M, TT) as a s e c t i o n ^ e T { U j n U^, ^*) of ^* over UjnUj,9^ 0 , {i;a forms a 1-cocycle with respect to U by (3.158): {fjk}eZ'{U, 0*). Suppose that each Uj is a coordinate poly disk. Then as with (3.160), the sequence

>H\Uj, €)^H\Uj,

€^)^H\Uj,Z)-^'

is exact. By Dolbeault's theorem (Theorem 3.12), we have H\ Uj, €) ^ n Uj, 5 ^ ) / 5 r ( Uj, si)

••

166

3. Differential Forms, Vector Bundles, Sheaves

where A is the sheaf of germs of C°° functions. By Dolbeault's lemma (Theorem 3.3), for any a-closed (0, l)-form (peT(Uj, dsd) on Uj, there is a il/er{Uj,sd) with (p=dil/, hence H\Uj,O) = 0. Since H^(Uj,Z) = 0, we obtain H\Uj, ^*) = 0. Therefore, by Theorem 3.5, we have H\n,

^*) = H\M,

^*).

(3.161)

Thus 1-cocycles {fjk} and {gjk}eZ^{Vi, 0^) belong to the same cohomology class if and only if there is a 0-cochain {hj} e C^(U, ^*) such that {fjkgjk} = 8{hj}, which means that^fcg7fc^ = M 7 ^ i-e. gjk = hj/n^hl^ Consequently {fj^} and {gjk} belong to the same cohomology class if and only if fji, and gjk are transition functions of the same line bundle F (see p. 99). Thus we identify a complex line bundle with the cohomology class of the l-cocycle {fjk}'F=

[{fjk}]eH\M,0%

Thus H^{M, ^*) is the group of complex line bundle over M with tensor product as its group multiplication. Definition 3.19. For a complex line bundle F e H\M, ^ * ) , S'^Fe H\M, Z ) is called its Chern class, and is denoted by c{F): c(F) = S'^F. Actually (3.161) holds for any sufficiently fine finite open covering 11 = {Uj}. In fact, take a finite open covering SB = { W^} of M such that each Wx is a polydisk. Then if each Uj is sufficiently small, we may assume that n = {Uj} is a refinement of SB: U<SB. Consequently by (3.71), (3.68) and Theorem 3.4, we have //^(SB, ^*) c: H\U, 6^*) c: H\M,

^*).

Since by (3.161), H H S B , ^*) = H{M, ^*), we have the desired result. Let U = { L/y} < SB be a finite open covering such that Uj and Uj n Uk (7^0) are all simply connected, and that t/jO UjnUk 9^0 is connected. Let F = [{fjk}]eH\M,€'^) where {fjk}eZ\n,€''). Then since ^ ( ^ ) ^y;^ is a non-vanishing holomorphic function on a simply connected open set Uj n Uk9^0, any branch of log fjk{z) is a one-valued holomorphic function on Uj n Uk. Put gjkiz) =

T= log fjkiz), 27rV-l

gkjiz) = -gjk(z).

Then by (3.158) e{gjk(z)-gik(z)

+ g,j{z))=fjk(z)fk{zr%(z)

= l.

§3.6. Complex Line Bundles

167

Hence Cijk = gjk(z) - gik(z) + gij{z)

is a constant integer on the doman UinUjn 11^9^0. Since e{gjk} = {yj^}, and 5{g,fc} = {Cyfc}GZ^(tl, Z), 5*F is represented by the 2-cocycle {Cy^} by the definition of 5* (see p. 130). Thus the Chern class ciF) = 8'^F of a complex line bundle F = [{fjk}] is given by

c{F) = [{c,j^}l

c,j^=

^ ( l o g X - f c - l o g ^ + logi;,-). 2W-1

(3.162)

By a divisor on a compact complei mdnifold M = M"" we mean a linear combination D = Y,^rn^S^ of a finite number of analytic hypersurface 5 i , . . . , S^,..., with coefficients m^eZ. For a sufficiently fine finite open covering 11 = { L^}, 5^ is given in U;, with L^ n 5^ ?^ 0 by its minimal equation f.j(z) = 0: S^ n (7,- = {z G ^ \f^j(z) = 0}.

(3.163)

In case L^ n 5^ = 0 , if we take as/^j(z) an arbitrary non-vanishing holomorphic function, e.g., a constant function 1, then (3.163) is also valid there. fvj{z)/fvk{z) is a non-vanishing holomorphic function on UjnUkT^ 0 . Put

fjk{z) = U['yY-,] vv/..(z); r

,

zG^-nf/,.

(3.164)

\jvk\Z)/

Then yjfc(^) is also non-vanishing on UjC^U^y^ 0 . Hence {fj^} G Z ^ ( U , ^ * ) . Thus there is a complex line bundle F with the transition function ^fc, which we call the complex line bundle defined by D and denote it by [D\.{D1^ = F = If S is non-singular and connected, 5 is a submanifold of M" of codimension 1, i.e. of dimension ( n - 1 ) . In this case we call D = S a non-singular prime divisor. A divisor D = Y,m^S^ on M is a (2n-2)-diinensional cycle, hence determines a homology class in H2n-2{^,^)- We shall investigate the relation between the Chern class c([D]) of [D] and the homology class of D. Consider a simplicial decomposition of M, and let / ? i , . . . , /7k,... be its vertices, Sj^.j< k, its 1-simplices, 5y/c, i<j< k, its 2-simplices, Sy^i, i <j
168

3. Differential Forms, Vector Bundles, Sheaves

Figure 4

For / < 7 < - • •
Since X m,;,-^^^^^ = ay = 0, and dstjj, = Sjj, - Sn, + 5^, we get I rriijkSjk -Z rriijkSik + 1 m^fc^y = 0,

(3.165)

which implies that (c, y) does not depend on the choice of the representative {Cijk} o f C.

For a divisor D on M = M"" considered as a (2n-2)-cycle, we denote the intersection number of D and y by I{D, y). Then we have I{D,y)

= {c([D]),y),

(3.166)

169

§3.6. Complex Line Bundles

To see (3.166), we assume for simplicity that D = 5 is a non-singular prime divisor and let fj{z) = 0 be a minimal equation of S on Uj. Since 5 is a submanifold of codimension 1, for any qe SWQ can choose local coordinates ( z ^ , . . . , Zq) centred at q such that fj(z) = zl Then for D = S, (3.164) reads

[5] is a line bundle with transition function yjfc(z). Assume that 5' meets no 1-simplices 5,^, and that it meets 2-simplices 5'^^ transversally if it does. Let I(S, Syj^) be the number of intersection points of 5 and s^^ counted with signature. Then HS,y) = T.^ijkHS,Sijk). Let Pijk be the centre of mass of 5^^ and Py the middle point of %, and decompose Syj^ into three cells ^yfc, e,fci, ^kij as illustrated below. If we assume that S does not meet any of three line segments joining p^^ with /Ty^, ptk and Pij respectively, then / ( 5 , s,j^) = I(S, e,j^) + I(S, ej^,) + / ( 5 , e^^.j).

Figure 5

170

3. Differential Forms, Vector Bundles, Sheaves

Since yj(z) = 0 is a minimal equation of S on Uj, and e^j^ c: Ut,

2W-I J,,,, by the residue theorem, where 36^^ is the boundary of 6^^. Hence

2W^/(5,5,,)= I

d log f,{z) +

d log fj{z)+\ 'jki

dlogA(z), ^ dej^ij

hence by (3.165) Pijk

dlogMz)~\

dlogMz) •f Pik

+ 1

d\ogfj{z)-\

+ Since fi{z)/fj{z) equal to

a

d log fj{z) •' Pa

'^ Pjk

JlogA(z)^ Pik

^logA(z)). ^ Pjk

'

=fij(z), and so on, the right-hand side of this equaUty is

Pijk

CPijk

diog^(z)Pjk

[Pijk

jiogy;,(z)+ ^ Pik

\

^iogi;,(z) ^ Pij

= 1 my7c(iogy;7c(Py/c)--iogi/fc(/'jfc)-iogy;fc(/7o7c) +iogy;fc(/),7c)+iogyy(Ajfc)-iog/y(/7y)). On the other hand, by (3.162), we have 10gX7c(/?yfc)-10g^(Py7c) + 1 0 g / y ( P y k ) = 2 7 7 y ^ C y f c ,

and by (3.165)

i;myfc(iogy;-fc(p,7c)-iogi;fc(A/c)+iogyy(Aj))=o. Hence / ( 5 , r ) = Im,^,c,, = (c([5]),r). (3.166) can also be proved similarly for general D.

'

§3.6. Complex Line Bundles

171

(b) Submanifolds of Codimension 1 Let 5 be a submanifold of codimension 1 of a compact complex manifold M = M" where we assume n^2. We denote by 0(-S) the sheaf over M of germs of holomorphic functions vanishing on S. €(-S) is a subsheaf of 0. We can identify €/(Oi-S)) with €s where Og is the sheaf of germs of holomorphic functions on S. In fact, let ( z ^ , . . . , z^) be local coordinates with centre p such that z^ = 0 on 5. Then we may regard ( z ^ , . . . , z^) as local coordinates on 5. In (0/€i-S)),= 0,/€(-S),, Op is the power series ring: Op = C{zp,..., Zp}. If we write (p{zp) e Op in the form oo

cp{zp) = cpo{zl,..,z;)+

I

(zir
we have (p{Zp) = (po(zl,..., z^) on S. Therefore 0(-S)p is the ideal of Op consisting of all power series of the form Z^^i (z^)'"
0(-S)p

= Op, and (O/O{-S))p 0/0{-S)

for pes.

= 0. Thus we may consider = Os.

(3.167)

The homomorphism rs:0'^Os = 0/0(-S) given by rs'. (p{zp) -» (po{z],,..., z ^ is called the restriction map to 5, because (Poizl, ...,Zp) = (^(0, z ^ , . . . , Zp) is the restriction of (p{zp) = (p(zl,..., Zp) to S. rs defines a homomorphism rgi T{W, 0)-^T(Sn W, Og) for any open set W^ M. For (peT{W, 0), rs(p is the local holomorphic function on S obtained by restricting (p io W r^S. Let U-{Uj} be a sufficiently fine finite open covering and ^(z) = 0 a minimal equation of S on Uj. Then we have 0(-S) = 0{[ST').

(3.168)

Proof. Let (p be a local section of ^(-5') defined over an open set W. In other words (p =
172

3. Differential Forms, Vector Bundles, Sheaves

Consequently on each Wn Uj 7^ 0 , (p{z) = fj{z)\jjj{z) with some local function i//r^(z) on Wn Uj. Since on Wn Ujn 11^9^0, fj(z)iljj{z) =fk{z)it/j,(z), we have iljj(z)=f;j!(z)Mz).

(3.169)

Consequently putting il/{z) = {z,il/j{z)) on each Wn Uj, we obtain a holomorphic section (/^: z^il/{z) of [5]"^ over W. Conversely, if if/: z^ il/{z) = (z, if/jiz)) is a holomorphic section of [S]~^ over W, ij/jiz) is a holomorphic function on Wn Uj such that^(z)(/fy(z)=/fc(z)(/^fc(z) on Wn Ujn Uk7^0. Consequently putting (p(z)=fj(z)il/j{z) on each Wn Uj7^0, we obtain a holomorphic function (p = (p{z) on W vanishing on S. Thus we obtain an isomorphism 0{-S) = 0([ST^). I The sequence

is exact, where t is the inclusion. Replacing 0(-S)

by €{[S]~^) we obtain

0 -> OUST') -^O^Os-^0.

(3.170)

Let F be a complex line bundle over M, €{F) the sheaf of germs of holomorphic sections of F, O(F-S) the sheaf of germs of holomorphic sections of F vanishing on S, and 0{F)s the sheaf of germs of holomorphic section of F over S. Then similarly as in (3.167) we have 0(F)/0{F-S)

= 0(F) s.

Let Fs be the line bundle over S obtained from F by restricting the base space to 5. We call Fs the restriction of F to S. Then 0(F)s is identified with ^(Fs): ^(F)5 = ^(Fs). Therefore we obtain as in (3.168) 0(F-

S) = ^ ( F ® [5]-').

(3.171)

Hence we obtain the exact sequence 0 -> 0(F®[ST')

-^ 0(F) ^ 0(Fs) -^ 0.

(3.172)

Putting F = [5], we obtain from (3.172) 0^O^O(F)^O(Fs)^0,F

= [Sl

(3.173)

since F®F~^ = M ® C . (3.172) holds also for holomorphic vector bundles of rank ^ 2 over M.

§3.6. Complex Line Bundles

173

Let K and K{S) be the canonical bundles of M and 5 respectively. Then we have K(S) = Ks®[S]s>

(3.174)

Proof. Let U = { L^} be a sufficiently fine finite open covering such that each Uj is a coordinate neighbourhood. Let (z],..., zj) be the local coordinates on Uj and fj(z) = 0 a minimal equation of S on Uj. Put fjk(z) =fj{z)/fk(z). Further assume that fj{z) = z] for UjnS9^0. Then ( z j , . . . , zj) may be regarded as local coordinates of 5 on 5 n Uj. The transition function of K is given by d{Zk, Zj^,...,

/^7c(^) = ciet 'd(z],z%...,z^y

Zfc)

Put J{S)j,{z) = dQt^}'l''''''ll

on

«9(z^-,... ,z^-j

5n^nt/,#0.

Then J(S)jk(z) is the transition function of K(S). Thus it suffices to prove J{S)j,,(z) = Jju{z)fjk{z) For 7, k with

on

SnUjnUk^0.

SnUjnUk7^0, Zk =fkj{z)z]

on

L^- n U^.

Hence dzl=fkj{z)

dz]-\-z] dfkj(z).

On the other hand, on Uj n Uf,, we have

J z j A dzl, A • • • A ^Zfc = d e t

2' '

' '

d{Zj,.

Since z] = 0 on 5, we have, on

n\ ^ ^ j ^ ^^7 A • • • A

dZj.

..,Zj)

SnUjnUj^

dzl A dzl A • • • A c/zfc =fkj(z) dz] A (izfc A • • • A dzl = fkj(z)J{S)jk{z)

Thus Jjkiz) =fkj(z)J(S)jk(z),

dz] Adz^A"'A

which proves (3.174).

I

dz].

3. Differential Forms, Vector Bundles, Sheaves

174

(c) Projective Space In this section we study complex line bundles on P". Let Zr,)3=i Scx^ dz"'®dz^ be a Hermitian metric on a complex manifold M", and o) =yJ-\ Za,/3=i ^«/8 ^^" ^ ^^^ ^^^ associated real (1, l)-form. We call "La 13=1 Sap dz'^®dz^ a Kdhler metric if d(o = 0. co is then called its Kdhler form. Any complex manifold M has a Hermitian metric, but does not always have a Kahler metric. A complex manifold endowed with a Kahler metric is called a Kdhler manifold. Let n^ be the sheaf of germs of holomorphic /7-forms over a complex manifold M = M". Put /z^'^ = dim H^(M, H^)

and

b, = dim H'(M, C),

where b^ is the rth Betti number of M. If M is Kahler, the following equalities hold: (3.175)

/z^'^ = /i^'^.

h'''' = br,

S

It follows from these equalities that br is even for odd r. Thus let M", n ^ 2 , be a Hopf manifold given in Example 2.9. M " is diffeomorphic to S^ x 5^"~\ hence cannot have a Kahler metric because its first Betti number is 1. If iV is a submanifold of a Kahler manifold M, restricting the Kahler metric of M to N, we obtain a Kahler metric of N. Thus a submanifold of a Kahler manifold is again Kahler. Let (^0? • • •» fn) he the homogeneous coordinates on P". Let U = { U,} be a finite open covering of P", {z],..., zj) local complex coordinates on Uj such that on each Uj, ^ ^ Q ) # 0 for some /3(j), and that

V ^J 9 • • • ' ^J /

, • • • 5

I

Xt^O)

^

fc/SO)

,

^

5 • • • J ^

(,13 U)

I•

^(3(j)/

Put a(z,) = l + Z L J z ; | l Then aa log a(z,)=

I

gj^pdzfAdzf,

where a(zy) The Hermitian form Xa^ gja^^^'vi^^ is easily seen to be positive definite where w \ . . . , w" e C . On the other hand, since ^(z^) =Za=o l^a/f^o)!^? we

§3.6. Complex Line Bundles

175

obtain a{zj) = \ejkiz)\^a(zk)

on

^0(7^7^0,

where ejk{z) = Cp{k)/Cpuy ^jki^) is a non-vanishing holomorphic function. Hence dd log a(Zj) = dd log a{z}^). Thus Y.a/3 Sjap dzf(S)dzf gives a Hermitian metric on P". Its associated real (1, l)-form is given by CO = V —1 dd l o g

a(Zj)

= V - 1 c/alog a{Zj), since J = (9 + 5. Therefore dco = 0, and X!« ^g gja^ dzf (x) dff is a Kahler metric onP". Thus P" is a Kahler manifold. Consequently projective algebraic manifolds are all Kahler, being submanifolds of the Kahler manifold P". Since the rth Betti number of P" is 1 for even r and 0 for odd r, where r ^ 2 n , we obtain from (3.175) dim H^P\

nn = h'^'^ = \ '

^^^'

py^q.

(3.176)

In particular for 0 = fl^, we have dim//^(P", ^) = 0,

q^l.

(3.177)

Hence by (3.160) we have H\P\

O"") = H\P\

Z) = Z.

(3.178)

Thus the group //^(P", 0^) of complex line bundles over P" is an infinite cyclic group. Let Poo be the hyperplane ^o = 0, and E =[Poo] the line bundle over P" defined by the divisor Poo- Since ColCpu)^^ is a minimal equation of Poo on Uj, the transition function of E is given by e,.(z)=-^/-^ =^ . inU)/

i>fi(k)

(3.179)

ipU)

E is a generator o/Z/^P", 0^). Proof. Let G be a generator of f/^P", (D^*), and suppose that E = G"^ where m is an integer. Let y be a line joining a

3. Differential Forms, Vector Bundles, Sheaves

176

point of Poo with ( 1 , 0 , . . . , 0). Then 7 = P M S a 2-cycle. Since c(E) = mc(G), we have, by (3.166),

I{P^,y) =

{c{E),y)=m{c{G),y).

On the other hand, /(Poo, r ) = U and {c(G), y) is an integer. Consequently m = ± l , and E = G or E = G~^. I Thus any complex line bundle Fe iir^(P", 0"^) is written as F = E^ where h is an integer. Let '^(^) = ' ^ ( ^ 0 , . . . , ^„) be a homogeneous polynomial of degree h in ^ 0 , . . . , L' Put ilfj(z) = ^(O/Cpij) on each Uj. Then iffjizj) is a holomorphic function of ( z ] , . . . , z") on Uj, and on Uy n L^^ ?^ 0 , ipjiz) = ejk(z)^il/k(zk). Therefore if/: z^ il/(z) = (z, i/(/(z)) gives a holomorphic section of JB^ over P": (/f 6 r ( P " , ^ ( £ ) ) where ij/jiz) is the fibre coordinate of the point (/^(z) of E\ We denote lA by il/ = {il/j{z)}, Conversely any holomorphic section i/^G r(P", ^ ( £ ^ ) ) with h a positive integer, is represented as il/= {ij/jiz)} with i//^(z) = '^(^)/^^(^) by some homogeneous polynomial ' ^ ( ^ of degree h in ^0, • • •, (n- Proof. Let Poo be the hyperplane ^o = 0. Then P" -Poo = C", and ( z i , . . . , z„) give complex coordinates on C", where z^ = CalCo- Since ^j{z) is holomorphic on Uj and since ^,(2) = ( y ^ )

^.(^)

on^nL/,^0,

we have on C" n Ujn 11^9^0

(fe)',«.(fe)V. (z). Therefore we can define a holomorphic function (p(zi,..., z„) on C" by putting 9 ( ^ 1 , . . . , ^n) = ( ^ )

^,-(z)

(3.180)

on each C" n ^ ^ 0 . Denote the homogeneous part of degree m of the power series expansion of cp by k. For this, take any point /7oo = (0, ^ 1 , . . . , f„) of Poo. Poo is contained in some Uj. If/?OOG ^ , we have, by (3.180), 00

yh

I j^'PMu...,L) = i'^aM^)m = 0 feo

(3-181)

§3.6. Complex Line Bundles

177

For a sufficiently small e > 0, the right-hand side of (3.181) is a holomorphic function in ^o in |^ol<e- Hence for m>/z, cpmUu" - An) = 0. Thus

is a homogeneous polynomial of degree Tz, and by (3.181) we obtain

Clearly r(P", ^(£^)) = r(P", 0) = C. For h<0, r(P", €(£'')) = 0 since in this case using (3.181) we obtain
u,-(z)i;(z) = e,,(z)V(^)A(^) on UjnUk9^0, which implies that {uj{z)fjiz)}eT(P",€{E'')). Since r(P", ^(E^)) = 0 for / i < 0 , and r(P", ^(JB^)) = C, h must be a positive integer. Consequently by the above result, {Uj(z)fj{z)} is represented as ^(^) u,(z)/;.(z)==^,

(3.182)

with a homogeneous polynomial ^ ( ^ ) of degree /z. On each Uj, fj(z) = 0 is equivalent to ^ ( f ) = 0. Thus 5 is a hypersurface of degree h defined by ^ ( ^ ) = 0. I Note that this is a special case of Chow's theorem (see p. 40). Let P{^) = P(^o, • • •, ^n) be a homogeneous polynomial of degree n ^ 2 in ^0, • • • 5 ^n- If at each point ^€ P", at least one of the partial derivatives dPiO/^L of PU) does not vanish, the hypersurface 5 defined by P(^) = 0 is non-singular. Moreover S is connected. For, if not, each component S„ I = 1, 2 , . . . , of >S is a submanifold of codimension 1 of P". Therefore by

3. Differential Forms, Vector Bundles, Sheaves

178

Theorem 3.23, S^ is a hypersurface defined by an algebraic equation ^ ( ^ ) = 0. Then ^ ( ^ ) does not vanish on 52. Therefore ^«/'^(^) with h = deg ^ ( f ) is a holomorphic function on 52, hence, a constant. Consequently c« = is also a constant, which implies that (fo, • • •, ^n) = (^o,..., c„) ia/^Uy^^ on 52. This is a contradiction. Thus 5 is connected, hence, is a submanifold of codimension 1. Let m = deg P(^), and put fj(0 = P{C)lC'^UY Thenyj(z) = 0 gives a minimal equation of 5 on (7, since at least one of the partial derivatives dP{^)/d^oc does not vanish at each point ^. Therefore fj{z)/fk{z) = {Cpi^k)/Cii{j)T gives the transition function of [5]. Thus [5] = The canonical bundle K of P" is given by (3.183)

X = E-"-\

Proo/ Let Uj = {^ G P" | {^ T^ 0}. Then U = {U,} is an open covering of P", and (z^,...,zi-\zi-^\...,z;)

z- = UCj

with

give local coordinates on Uj. Define Jj^ by d z l ^ - ' ^

dzt'

A J Z ^ ' A • • • A J Z ^ = J^-fc6?Z,% • • • A J z j - ' A J z j " ' ' A • • • A J z ;

on Uj n Uk' Then K is defined by {Jju}- On the other hand, the transition function Cj^ of E is given by ey^ = Ckld- Since z^ = ZQ/ZQ, a = 1 , . . . , n, and Zfc= l / z j , we have dzl^dzl^'

' ' ^ dzt~^ A dzt^^ N- - Kdzl

/ ^ \ n+l dZoAdzlA-'-AdZo^A = -( — 1

dZo^'^ A'

' '

AdZo

^ 4 A • • • A dz!^-' A ^z5 A • • • A dzl

= (-1)''(—)

T h u s / o . - ( - l ) ' ^ ( f o / f J " ^ \ hence

Consequently K = £'"""\

|

(d) Surfaces in P^ Let 5 be a non-singular surface of degree h in P^. By the results of the preceding subsection, [S] = E^ where E = [P^].

§3.6. Complex Line Bundles

179

By (3.183) K = E-^ for ¥>\ Therefore by (3.174) the canonical bundle K{S) of 5 is given by K{S) = E'r\

(3.184)

where Es is the restriction of E to S. The first Betti number b^ of S vanishes. In fact we will let Cl Co~'Cu . . . , CHHHe\ . . . , CiCt'. Cs be all the monomials of degree h in ^0, ^1, ^2, ^3- They are (^3^) in number. Using these monomials we define a map of P^ into P"" where JJL = Ct^) - 1 , as follows: L: ^-> (COO, WJ, . . . , ft>^) = (^5, ^ o " Y i , . • . , ^3*),

where (a>o, W i , . . . , co^) is the homogeneous coordinates of P^. t maps P^ biholomorphically into P ^ Thus P^ = 6(P^)cip^ is a submanifold of P ^ Let ^ ( ^ ) = 0 be the defining equation of S. ^ ( ^ ) is a linear combination of monomials of degree h\ n n = aoCo + «i^o ~'^i + • • • + a.^oVf ^^2^^ 3'^+ • • • + a^^,\ Therefore if / / is a hyperplane of P^ defined by ao<^o+ • * • + ci^^o)^ — 0, then ^ ^ p 3 ^ p M -g ^j^^ section of P^ by H: / / n P ^ = 5. Consequently by Lefschetz' theorem (see [23], also [24] for the proof using Morse theory),

H\S,0

=

H\P\0.

Since / f ' ( P ' , C) = 0, bj = 0 for S, Let nVbe the sheaf of germs of holomorphic 1-forms on S. Since 5 is Kahler, and h^ = 0, we obtain from (3.175) //'^(5,a') = 0.

(3.185)

Let M be a 2-dimensional compact complex manifold which is not necessarily Kahler. Then we can prove that if //^(M, C) = 0, then /f ^(M, a ' ) = 0 as follows. Let

|(/^i2| ^z^ A dz^ A Jz^ A Jz^

JM

—~ J M

J M

l'Ai2|^ dz^ ^ dz^ ^ ^^^ ^ dz^

3. Differential Forms, Vector Bundles, Sheaves

180

On the other hand,

dcp A d(p= \ JM

d(p Ad(p= \

d{(p A d(p) = 0.

JM

JM

Hence (/^i2 = 0. Thus dcp — 0. Since by hypothesis H^{M, C) = 0, by de Rham's theorem r ( M , dsd^) = dr(M, ^''). Therefore for any ^ e r ( M , dsd""), there i s a n / e r ( M , si"") such is a (0, l)-form, df=0. Therefore / is that (p = df. Since (p = df=df+df holomorphic on M, hence a constant. Consequently (p = df=0. Thus we obtain H%M,a') = 0. If dim M = 2, H\M, Z ) - Z. Therefore the cup product cc' e H\M, Z ) = Z of any two cohomology classes c, c' G H^{M, Z ) is considered as an integer. In general, for a compact complex manifold M = M"^, the ^r^^ Chern class Cj of M is defined to be —c{K) where K is the canonical bundle of M: Ci = —c{K). For n = 2, c\= CiCi is an integer. In this case we call (l + Cj) the linear genus of M. Let 5 be a non-singular hypersurface of degree h in P^. Then c{Esf = h.

(3.186)

Proof, Let P be a plane in general position in P^. Then P is not tangent to 5', so the intersection C = P n 5 is a non-singular algebraic curve on 5. C, as a divisor on S, defines a line bundle [C] on 5. Then we have £s = [C].

(3.187)

In fact, let /(^) = 0 be the defining equation of P where 1{C) is a linear form in fo, Cu ^2, ^3- Let U = {Uj} he a. finite open covering of P^ such that on each Uj, some ^^Q)7^0. Then KOIC^{j)='^ gives a minimal equation of P on Uj. Therefore on 5 n U, T^ 0 , il{0/Cf3(j))s = 0 gives a minimal equation of C = P n 5, where {l(0/Cp(j))s denotes the restriction of 1(0/^^u) ^^ '^• Consequently the transition function of [C] is given by

Since ^y^ = ^/3(k)/^/30) is the transition function of E, we obtain [C] = E5 as desired. Let P' be another plane in general position in P^, and put C' = P' nS. Then C and C meet transversally on S. Then

c([C])-c([C']) = /(C,C'),

§3.6. Complex Line Bundles

181

where / ( C , C ) is the intersection number of C and C (see [15]). Since by (3.187), £s = [C] = [ C I , c{Es)' =

I{C,C'),

On the other hand, y = PnP' is a line in P^, and the intersection points of C = P n 5 with C = P' nS coincide with those of S with y. Since a line 7 in general position meets a hypersurface of degree h transversally in h distinct points, / ( C , C) - h, hence c{EsY = h. I

Chapter 4

Infinitesimal Deformation From now on we proceed to the main theme of this book.

§4.1. Differentiable Family (a) Introduction As stated in §2.1(a) in connexion with Definition 2.1 of complex manifolds, a complex manifold M = M " is obtained by glueing domains ^i,.,.,%,... in C": Af = U j %. If we write the coordinate transformation by yj^ instead of Tjj, in (2.2),

ffj,: zj, -> zj = {z],...,

zj) =fjkizk),

fji^ is a biholomorphic map of the open set ^kj ^ ^k onto %k ^ %, and Zje% and Zf^e^L^ are the same point of M if Zj=fjk(z},). Zj==fj,^{z,,) is written in detail as Zj =f%{zk) =mzl...,

zD,

a = 1 , . . . , n.

(4.1)

In the sequel we write Uf instead of %y for simplicity: Uj = %. By Theorem 2.1, we may assume that U = {^ |j = 1, 2,...} is a locally finite open covering of M, and that each Uj is a polydisk: t/, = { z , e C " | | z j | < r ; , . . . , | z ; | < r ; } .

(4.2)

Replacing zf by zj/rf, we may assume further that Uj is of radius 1: [/, = { z , e C " | | 2 ] | < l , . . . , | z ; | < l } .

(4.3)

If M is compact, U = { Uj} may be assumed to be a finite open covering. Thus a compact complex manifold M is obtained by glueing a finite number of polydisks U^ ..., Uj,..., Uj by identifying z^ e Uk and Zj =fjk(zk) e Uj'. M = Uj-=i ^j' ^ deformation of M is considered to be the glueing of the same polydisks Uj via different identification. In other words, replacing

§4.1. Differentiable Family

183

fjk(zk) in (4.1) by the functions fjkiz,, t) =ff,(z,,

?!,..., tj,

ff,(z^, 0)

=ff,iz,),

of Zfc and the parameter t = (ti,..., t^), we obtain deformations M^ of M = Mo by glueing the polydisks Ui,..., Ui by identifying z^ G U^ with z^ =fjkizk, t)e Uj. This is the fundamental idea of Spencer and the author on the deformation of complex structures [21]. We were, however, at first rather sceptic about this idea. For, the complex structure of M^ = U^. Uj thus defined via the identification Zj=fjk(zk, t) may not actually vary with t. For example, let ^^ = ( ^ j , . . . , ^J) be local coordinates of M, and choose holomorphic functions zj = zfi^p 0, a; == 1 , . . . , n, of (« + 1) variables Up t) = Ul..., ^^j), defined on (, e Uj, |r| < 1, such that the map Zj'. ^j-^ZjUj. t) = (z'jUj, 0, . • • , Z^Uj, t)) gives new local coordinates of M on Uj. Let Zk-^Zj=fjk{Zk,

t)

be the coordinate transformation. Then since

ffki^k, t) depend in general on t. But since {z^,..., z,,...} is a system of local complex coordinates of the given complex manifold M, M^ = U^. Uj obtained via the identification Zj^fjk{z^, t) is just M, hence M^ does not vary with t. Thus at a glance M^ appears to depend on t, but actually does not. Therefore since, at first, we had no criterion for the acutual dependence of Mf on t, we were sceptic about the efficiency of our above-mentioned idea. Later at a colloquium held in the Institute for Advanced Studies at Princeton, A. Frolicher reported his joint work with A. Nijenhuis on the deformation of complex structures, and explained their proof of the rigidity of P" ([7]). Their theory depended on the differential geometric method developed by them. This encouraged us to develop our idea on the deformation anew. (b) Differentiable Family At first we did not take notice of the fact that the parameter appeared in the definition of a complex manifold is in general a complex one, and that therefore as a family of complex manifolds M^ depending on the parameter t = (ti,..., t^), it is more natural to consider a complex analytic family (Definition 2.8 in §2.3). Thus we at first considered the case that/j^(zfc, t) =

184

4. Infinitesimal Deformation

fjki^k, ti,... ,tm) are C°° functions of the real parameters ti,..., t^. For this we introduced the concept of a differentiable family of compact complex manifolds. The definition of a differentiable family of compact complex manifolds is the C°° analogue of the definition of a complex analytic family {Mt\te B} = {M, B, m) given in §2.3(a). We give the precise definition below. Let J^ be a differentiable manifold, B a domain of IR'", and m a C°° map of M onto B. Suppose that the rank of the Jacobian matrix of xu is equal to m at every point of Ji. Then, as stated in §2.3(a), we can choose a system of local C°° coordinates { z i , . . . , x^,...}, xy. p^ Xj(p) satisfying the following conditions: (i) (ii)

Xjip) = (xj(/7),..., xj(p), r i , . . . , tj, (tu ...,tm) = m(p), {%} is a locally finite open covering of M where % is the domain o f Xj.

Therefore for every point te B,if m'^t) is connected, Trr'^t) is a differentiable manifold whose system of local C°° coordinates is given by {P^(x](p),...,xj{p)\%nm-\t)r^0}. Moreover if rxT'^t) is compact, taking for each feB an open multi-interval / such that r ^ G / c i [ / ] c : B , we see from Theorem 2.4 that VT~\I) =

m-\f)xL Definition 4.1. Suppose given a compact complex manifold M^ = M " for each point r of a domain B of IR'^. {M, \teB}\s called a differentiable family of compact complex manifolds if there are a differentiable manifold M and a C°° map m of M onto B satisfying the following conditions: (i)

The rank of the Jacobian matrix of m is equal to m at every point of M. (ii) For each teB, 'UT~^{t) is a compact connected subset of M. (iii) m-\t)^M,. (iv) There are a locally finite open covering {% |j = 1, 2,...} of J^ and complex-valued C°° functions z j ( / ? ) , . . . , z]{p), j = 1, 2 , . . . , defined on %ij such that for each t {p-^{z]{p),...,2l{p)\%n^-\t)^0}

(4.4)

form a system of local complex coordinates of M,. By (i), (ii), each m~^{t) is a compact differentiable manifold. The condition (iii) means that m~\t) is the underlying differentiable manifold of M^. We call teB the parameter of the differentiable family {Mj rG B) and B its parameter space or base space.

§4.1. Differentiable Family

185

We denote the differentiable family {Mt\tG B} by (Jt, B, m) or simply by M. If we write zJ{p) = x]"'~^{p)-\-ix]'^{p), a == 1 , . . . , n, by dividing into the real and imaginary parts, and put ^jip) = ix]{p), xj(p),.

. ., x f (/7), t,,..., tj,

{t„ ...,tj

= m{p),

then by the condition (iv), {Xj I7 = 1, 2 , . . . } , xf. p -^ Xj{p) form a system of local C°° coordinates of the differentiable manifold M. Xj maps % into R^" X B differentiably. Introducing complex coordinates in R^" by (z\ . . . , z") = (x^ + ix\ . . . , x^"-' + ix^"), we consider !R^" as C", and Xj as a C°° map of % into C" x J5: Xj: p -» x,(;?) = {z]ip),...,

z;(;7), ^ i , . . . , r^),

(t,, ...,tj

= m{p).

(4.5)

When we consider x, as a map of % into C" x J5, we call {xj I7 = 1, 2,...} a system of local coordinates of the differentiable family Ji. Since the coordinate transformation Xk(p)^Xj(p) for % n ^^ ^^ 0 does not affect ^ 1 , . . . , ^m, it is written in the form izlip\...,zUp\t)^iz'j(p\...,z^(p\t)

= (fjkizkip, t), r),... Jjkizj^ip, t), t), r),

t = m{p).

Since by (iv), for a fixed t,p-^{z]{p),... ,z]{p)) gives local complex coordinates of the complex manifold M^, the coordinate transformation on

{zi{p),...,zl{p))^{z]{p),...,z]{p))

is biholomorphic. Therefore C°° functions fU^k. t) =fU^l

...,zl,t,,...,

tj,

a = 1 , . . . , n,

of Zfc,..., Zfc, ti,..., t^ are holomorphic in z ^ , . . . , z^. There are infinitely many choices of systems of local coordinates of the given differentiable family Jt. Given a locally finite open covering {W;^\X = 1,2,...}, and a diffeomorphism u^:p-^u^{p)

= ( w l ( / ? ) , . . . , wlip), ^ 1 , . . . , tm),

of W), onto an open subset of C" x B for each A, {MJ, . . . , M^, • • •} form a system of local coordinates of the differentiable family M if and only if for

186

4. Infinitesimal Deformation

each t e B, w^ip), with pe MjnW^ is a. local holomorphic function on M^. Thus if {MA} is a system of local coordinates of the family Ji, putting

<(P) = gtj(z]{p),...,

zjip), ^ 1 , . . . , tj,

ih, ...,tj

= m(p),

on each W^ n %, we see that the C°° functions gtjizl...,

z;, ^ 1 , . . . , tj,

a = 1, 2 , . . . , n,

(4.6)

of z ] , . . . , z j , ^ 1 , . . . , tm, are holomorphic in z ] , . . . , zj. Given a differentiable family {Jt, B,m), by Theorem 2.1, we may choose its system of local coordinates {Xj}, xf. p -» Xjip), Xj(p) = (zj(/7),..., z;(/7), ^ 1 , . . . , tm),

{tu ...,tj

= m{p),

such that for each j w e have Xj{%)=UjXlj,

(4.7)

where we put ^ = { Z , G C " | | Z 7 | < r 7 , a = 1 , . . . , n}, and Ij = {teB\aj^< K<^jir> ^ = Ij • • •, ^ } - Thus identifying p e % with Xj{p), we consider ^ = [JJ Uj X 7^-. From this viewpoint, (z^, t)e UjX Ij and (z^, t)e U^x 4 are the same point on Ji if Zj=fjk(zjc, t). In terms of the local coordinates (zj, r), tar is written as t i T : ( z j , . . . , z ; , o->^. Consequently m{UjXlj) = Ij. Thus M^ = tz7~^(0 is the union of all UjXt with r G Ij, Identifying Uj x t with Uj, we may consider M,= U Uj. IjBt

Then z^ e L^ and z^ e Uj^ are the same point on M, if z^ =fjk(zk, 0- Thus M^ f5 a compact manifold obtained by glueingpolydisks Uj with IjStby identifying Zfc e Uk with Zj =fjk(zk, 0 e Uj. Since {% IJ = 1,2,...} is locally finite, for any fixed t^e B, there are only finitely many % such that % n M,o 7^ 0 . For simplicity we put t^ = 0, and assume that %nMo7^0 forj = 1 , . . . , / , and that % n MQ7^ 0 fovj^ / + 1. Then MoCzU UjXij.

187

§4.1. Differentiable Family

Putting I = r)j=i Ij, and Mi = m \l),

we have

Mr = m-\I)

= [J

UjXl.

(4.8)

j=i

UjXl

M

U,xl

Figure 1

Of course / is an open multi-interval with Oe I(^ B. Therefore for each te I, Mt = U^^i Uj is a complex manifold obtained by glueing Ui,..., Ui by identifying Zj, e Uj, with Zj =fjk{zk, t) e Uj. Thus considering a differentiable family (Jt, B,m),wQ can get a clearer image of the deformation of compact complex manifolds. Namely, given a compact complex manifold M, if there is a differentiable family (Ji, B, wr) with m~\t^) = M for some t^eB, each Mt = m~\t) is called a deformation of M By Theorem 2.4, ifMt is a deformation ofM, M, and M are diffeomorphic. Restricting the domain of the parameter t to a sufficiently small multi-interval /, we see that M, = V7~\t) = U^.^i ^ for t e /, where each Uj isapolydisk independent oft, and only the coordinate transformations z^ -^ Zj = fjk(zk, t) depend on t. Thus only the way of glueing Ui,..., Ui depends on t. Each point of M^ belongs to one of Uj. Consequently the complex structure of a sufficiently small neighbourhood of each point of M^ does not vary under deformation. Note that at this point the deformation of complex manifolds is essentially different from the deformation of an elastic body. For, any small portion of an elastic body undergoes distortion under deformation. Let / 7 G M , = UJ=i ^7,, and suppose peUinUjnUk. Then putting Zi = Ziip), Zj = Zj(p), and z^ = Zk(p), we have Zi =MZj,

t) =fkiZf,,

t),

Zj =fjk{Zk,

t).

188

4. Infinitesimal Deformation

Hence on % n % n^k9^0,

we obtain the following important equality.

Mz,,t)=Mfjj,(z^,t),t).

§4.2.

(4.9)

Infinitesimal Deformation

(a) Infinitesimal Deformation Given a differentiable family (Jt, B, wr) of compact complex manifolds, how can we know whether the complex structure of Mt = vT~\t) actually depends on tl First let us consider this problem for m = 1, that is, in case B is an open interval of IR. Given a differentiable function f{t) of a real variable t, by differentiating it in t, we know whether/(O is a constant independent of t or not. Analogically also in our case we expected to obtain a useful criterion of the actual dependence on t by the differentiation of (4.9) in t. (4.9) is written in more detail as

fUz,, t) =mfjic(zk, 0 , . . . JMz^, 0, 0,

a = 1,..., n.

Putting zf =ffj^(zk, t), we obtain

dmzj,,t)_dfrj{zj,t) ^ " dftjizj,t) df%{zu.t) dt

dt

(3 = 1

dzf

dt

Also putting zf =fy{zj, t), we have

dniz(,,t)_dfrj(zj,t) ^ ^ dzidfixzj^ dt

dt

13 = 1 dzf

dt

Using holomorphic vector fields, we can rewrite these equalities in the following intrinsic form: J dftkizk^t) a=i

dt

d _ " dftjizj^t) dzf

a=i "

dt

dt

where we use the equality d

"

dzf

dzf ~ hx dzf

dzf

df%{Zk,t)

%t,

d dzf

d

d

dzf

§4.2. Infinitesimal Deformation

189

Introducing the vector fields OjM = i ^ ^ ^ ^ - ^ , a=l

Ot

z, =A,(z,, t),

(4.10)

OZj

we write the above equality as eidt) = o,j(t) + ej,,{t).

(4.11)

Thus on each open set Ujn 17^7^0 of Mt = [Jj Uj, a holomorphic vector field Ojk(t) is defined, and the equality (4.11) holds on UinUjn Uk^0. (4.11) is also written as ^,fc(0-^.fc(0 + %(0 = 0.

(4.12)

Putting i = k, since fZk = ^k, we have dkk(t) = 0. Consequently from (4.11) we have ekj(t) = -ejM.

(4.13)

From (4.12) and (4.13), we see that {Sj^it)} is a 1-cocycle. Namely, let St be the sheaf of germs of holomorphic vector fields over M^. Then ^,-,(r)Gr(^-n(7,,©,),

Otj(t) = -ejM.

From (4.12), {Ojj^it)} is a 1-cocycle with respect to the open covering Ut = {Uj} on M, = U , Uj' {Ojk(t)} e Z^U,, 0,). We denote by e{t) the element of the cohomology group H\Mt, St) determined by the l-cocycle {djj^{t)}. By the Corollary to Theorem 3.4, H^{Vit,St) is a subgroup of H\Mt, St): H\nt,St)^H\Mt,St).

(4.14)

Thus 6(t) is considered to be an element of H\Mt,St). Then it will be obvious that 0{t) is the cohomology class of {6jk(t)}. Since Ojk(t)=

I a^l

— , ot

Zk=fkj(zj,t),

dZj

is obtained by differentiating fjki^k, 0 which define the complex structure of M^f, we may expect that 6{t) represents in some sense or another the "derivative" of the complex structure of M^ with respect to t. Hence we call 6{t) the infinitesimal deformation of Mt, and consider 6(t) as the derivative of the complex structure of Mt with respect to t. We denote it by

190

4. Infinitesimal Deformation

dMJdv.

f =.(-).

(4.,5,

In the above argument, we assume that Xj{%) = UjX Ij, but this assumption is redundant. Let { x i , . . . , Xj,...} be an arbitrary system of local coordinates of the differentiable family M, and % the domain of x,. Identifying p e % with (Zj, t) = (z],..., Zj,ti,..., tm) = Xj{p), we consider % with {Zj, t) = C " x B , hence %nMt=Uj,xt,

Uj,^C\

Consequently identifying Ujt x t with Up, we obtain an open covering Vit = {Ujt} of Mt. Ut is a finite covering of M^ where we omit Up if it is empty. Let {Zj,,t)-^{Zj,t)

=

{fj,,(Z},,t),t)

be the coordinate transformation on % n ^k. Then we have the identity (4.9) on %n %n^j,9^0. Consequently defining Oj^it) by (4.10), we have {^jk{t)} e Z\Vit, ©f). We denote by 0(t) the cohomology class of the 1-cocycle {Ojk(t)}:

and define ^ ( 0 as the infinitesimal deformation of Mt'. dMt/dt = 0{t). Thus the infinitesimal deformation is defined with respect to any system of local coordinates. We must verify that d{t) does not depend on the choice of systems of local coordinates. First we show that ^ ( 0 does not change under the refinement of U = {%}. Let 93 = { r , } , r^ c: ^^.(^), be any refinement of U, and define local coordinates x^ on Y^ by x^,: p^X;,{p) = Xj(^^^(p). We must show that the infinitesimal deformation defined with respect to {x} coincides with ^ ( 0 defined with respect to {Xj}. Since X;, is the restriction of x^(x) to T^, the holomorphic vector field ^AAX(0 defined by (4.10) with respect to {x^}, is the restriction of Ojixmtx) to T^nV^n m'^t). Namely, putting 93, = {T^J, where TA, = "FA n M, T^ 0 , we have

Hence {0),fj,(t)} = U.^\{djj^(t)}. Consequently by (3.81) the cohomology class of {^A/Lt(0} coincides with that of {Ojk(t)}. Thus the infinitesimal deformation does not change under the refinement of the open covering. Since any two locally finite open coverings have a common refinement, in order to prove the independence of ^ ( 0 of the choice of systems of local coordinates, it

§4.2. Infinitesimal Deformation

191

remains to show the following: Given two local coordinates Xj = (zj, t) and Uj = {Wj, t) on each Uj, the infinitesimal deformation 7]{t) with respect to {Uj} coincides with ^ ( 0 defined with respect to {Xj}. Let (Wfc, t) -> {Wj, t) = (hjj,(Wk, t), t)

be the coordinate transformation of {Uj} on % n ^^ T^ 0 . Put as in (4.10) Vjk{t)=

I

—^,

Wk =

hkj{wj,t).

Then 7){t) is the cohomology class of the 1-cocycle By (4.6),

{r)jk{t)}eZ^{^t,%t).

w; = gf(z;,...,zr,o is a C°° function of z],...,

z", t, which is holomorphic in Zj,...,

z". Since

gf(zj, t) = w; = hf^iwk, t) = hfkig^izk, t), t), and Zj =fjkizk,t),vje

have gfifjki^k, t), t) = hjkigkizk, t), t)

(4.16)

on '^k <^ % ^ 0- Differentiating (4.16) in t, we obtain « dgldj% p^l dzf

dt

p = l dt

dzf

dgl_ dt

"

dh]i^dgld_h%

i^idw^

dt

dt

that is, dt

13 = 1 dt dW^

dt

Multiplying d/dwj from the right, and taking the summation Sa = i» we obtain the equality

p = l dt

dzf

a = i dt

dwf

phi

Putting a=l

ot

dWj

dt

dW^

a=l

dt

dwJ"

192

4. Infinitesimal Deformation

we have Ojdt)-Vjk(t)

= OUt)-ej{t).

Since Oj(t) is a holomorphic vector field on Ujt = %nMt, C\U,, e,) and (4.17) means that {0Mt)}-{vj,(t)}

(4.17) {dj{t)}e

= 8{ejit)}.

Therefore r](t) coincides with 6(t), and the infinitesimal deformation 0(t) does not depend on the choice of systems of local coordinates. (b) Trivial Differentiable Family The first problem in the theory of deformation was to see if we could justify the consideration that the infinitesimal deformation d(t) represents the derivative of the complex structure of M^ with respect to t For, this consideration seemed to the author too good to be true. (Spencer had, I think, a more optimistic view.) If 6(t) is the derivative of the complex structure of M^ with respect to t, 6{t) = 0 must hold if Mf does not vary with t. For this we must give the precise definition for M^ not varying with t. In general we define the equivalence of two differentiable families as follows. Definition 4.2. Suppose given two differentiable families {M, B,m) and {Jf, J5, TT) with the same base space BaW, M and M' are called equivalent if there is a diffeomorphism ^ oi M onto ^ such that for each teB, <^ maps Mt = m~^{t) biholomorphically onto Nt = 7r~\t). Suppose that Ji and ^ are equivalent. Let {Xj}, xy. p -> Xj{p) = (zj(p), t), t = m{p),hQ the system of local coordinates of Jt, and {ux}, U;^: q -^ U;^{q) = ( ^ A ( ^ ) , OJ ^= '"'(^)5 that of ^ . Then we have u,(^(p))

=

{w,(^(p)),t),

If we represent w"(0(P)) by a C°° function of the local coordinates (zj{p), t) as

then gtizj, t) is holomorphic in z],... ,z]. In this case, identifying peM with ^ = (/7) G ^ , we may consider that M and ^ are the same differentiable family. In fact, since 4> is diffeomorphic, we may identify ^ and J^ via O. Then since gti^p 0 is holomorphic in z],..., zj, the system of local coordinates {WA}, "A = {^k, t), ^\ = S\(^j^ 0, of -^ can be regarded as a system of

§4.2. Infinitesimal Deformation

193

local coordinates of M. Thus if M and Jf are equivalent, we may consider Jf as the same differentiahle family with M endowed with a new system of local coordinates. Let M be a compact complex manifold, and {WA} a system of local complex coordinates. Let TT: M x B ^ B be the projection of M x B onto B. Then (M x 5, B, TT) is clearly a differentiahle family of compact complex manifolds. A system of local coordinates of M x B is given by {WA}, where "A = {^K, 0- Of course the complex structure of TT~^{t) = M xt = M does not depend on t. Definition 4.3. A differentiahle family (M, B, m) is called trivial if it is equivalent to (MxB,B, IT) where M = m~\t^) with t^eB. If the differentiahle family {M, B, m) is trivial, M coincides with M xB endowed with some system of local coordinates {Xj}, Xj = (zj, t) = (Zj,..., z", t). Let %LjCi M xB be the domain of Xj, and {w^} a system of local complex coordinates of M = m~^{t^). Then z7 = g7(wA, t),

a = l,...,n,

are C^ functions of w ^ , . . . , w", r defined on %Lj which are holomorphic in w\,.,,,wl.On%n^^,

z^=ff,(z,,t) are C°° functions of z j , . . . , Zj,t which are holomorphic in Zj,..., z". For any fixed t, {Zj \%n M xt} form a system of local complex coordinates of Mt = m~\t). Since in general the coordinate transformation Zj^'^Zj = fjki^k, t) does depend on /, the complex structure of M^ apparently depends on t. Let / c : B be a subdomain of B. Then {Mj, /, m) with Mj = vj~\l) is clearly a differentiahle family, where the domain of vr is assumed to be restricted to Mj. Definition 4.4. A differentiahle family (M, B,m) is called locally trivial if for each te B, there is a subdomain / with telci B such that {Mi, I, m) is trivial. By saying that the complex structure of Mt = V7~\t) of the differentiahle family {Ji, B, m) does not vary with t, we mean that {M, B, m) is locally trivial. If (Ji, B, m) is locally trivial, each M^ = tiT~\t) is biholomorphically equivalent to a fixed M = m'~\t^). Let us consider the inverse. Namely, if each M^ is biholomorphic to a fixed M, is (M, B, w) locally trivial? This problem was solved for complex analytic families affirmatively by Fischer and Grauert [4] later. (This paper

194

4. Infinitesimal Deformation

was published in 1965, but Grauert explained the idea of proof already about 1960 at the "Nothing Seminar" held in Princeton University under the direction of Spencer.) At that time, however, nothing was known about it. We thought that the local triviality of {M, B, m) would be stronger than the condition that each M^ is biholomorphic to a fixed M Now again let B be an open interval of R. If {M, J5, m) is locally trivial, for any sufficiently small domain / c : J5, {Mj, /, m) is equivalent to {M xl, /, TT). Since the infinitesimal deformation d(t) of Mt = m~^(t) does not depend on the choice of systems of local coordinates, we can calculate d{t) for tel in terms of local coordinates {MA}, WA = (^A, t) of M X /, where {w^} is a system of local coordinates of M. Let Wf^^Wx = /IA^ (n^^Lt) be the coordinate transformation for {w^}. Then the coordinate transformation for {1/^} is given by (w^, t)-> (WA, 0 = (h^(^^),

t).

Since hx^(w^) is independent of t, we have

hence ^ ( 0 = 0. Thus if the complex structure of M^ = rjiT'^t) does not vary with t in the sense that (M, B, m) is locally trivial, we have 0(t) = 0. IfO(t) is truly the derivative of the complex structure of Mt with respect to t, that 6(t)=0 identically must imply conversely the local triviality of (M, B,m). We shall show below that this is true at least under a certain additional condition. (Cf. [21], Chap. II.) For simplicity we consider the differentiable family Mi = m~\l)

/ = \J

UjXl

7=1

given by (4.8). Each Uj ci C" is a polydisk, and (z^, t)eUjXl UkXl are the same point on Jij if z"=f]ic(Zk,t),

and (z^, t) e

a = l,...,n.

Let St be the sheaf of germs of holomorphic vector fields on M^ = w~\t), and nt = {Uj} = {UjXt}. Suppose ^ ( 0 = 0 identically. 6(t) is, by definition, the cohomology class of the 1-cocycle {^,,(0}eZHU., ®.) where Ojf,{t)=H^,(df%/dt)(d/dz^). Since by (4.14),

e{t) e H'iUt, e,) = z\Ut, e,)/5C«(U„ ©,),

§4.2. Infinitesimal Deformation

195

^ ( 0 = 0 means that {Ojj^(t)} = 8{dj(t)}, namely there is a 0-cochain {6j(t)}e C ^ U „ e , ) such that Ojk(t) = ek{t)-ej(t).

(4.18)

We want to prove that {Mi,I,m) is equivalent to ( M x / , / , TT) with M = m~\0). In other words, we must show that there is a diffeomorphism 0 of M X / onto MI = U]=i UjXl such that O maps M xl biholomorphically onto Mt = m'^t) for each t We proceed as in the proof of Theorem 2.4(1°). Let Oj(t)=i

O^zj^t)^, a=l

OZj

Then the equality (4.18) is written as

a =l

Ot

dZj

a=l

OZk

OZj

a=l

We denote by {d/dt)u the vector field d/dt on UkXla Mj. Then with respect to the coordinate transformation (z^, t) -> (Zj, t), zf ^ffki^k, t), a = l,... ,n, we have as in (2.26)

\dt)k

«=i

dt

dzJ

Consequently on each open subset UjXln

a=l

dZj

\dt/j

\dt)j'

1/^x19^0

a^l

dZk

We define the vector field v on Mj, by putting on each v=- l

a=l

0^izj,t)^+^^. dZf ^Zj

of Jij we have

dt

\ot/k

UjXl (4.19)

(4.19) is an analogy of (2.25). But in contrast to (2.25), it does not follow immediately that v is C°°. In fact, though 6jk(t) is a C°° vector field on UjXln UkXl^ Oj(t) may not be differentiable in t. Moreover 6j(t) is not uniquely determined by {Ojk{t)}. Hence there arises the following problem: Can we choose dj{t)J =\,... ,1, such that each 6f(Zj,t) is C°° in z],..., zj,tl The following theorem is an answer to this problem. Theorem 4.1. If dim H\Mt,

St) is independent of te I, then we can choose

196

4. Infinitesimal Deformation

a 0-cochain {Ojit)} with 8{0j{t)} = {Oj^it)} such that each ej{Zj, t) is a C°° function of z],..., z", t where we put Oj{t) = YZ = \ ^Ji^p t)(d/dzf). We give the proof of this theorem in §7.2(b). The proof depends on the method developed by Spencer and the author for the theory of variations of almost complex structures ([19]). Now suppose that dim H\Mt, 6 J does not depend on t. By Theorem 4.1, choose {6j{t)} such that each dj(t) is a C°° vector field on Uj xlaJli. Then v in (4.19) is a C°° vector field on Jij. Consider the simultaneous ordinary differential equations dzf

1

a = l, (4.20)

dl = 1, ds for V as in (2.28). Take an arbitrarypeM = m ^(0), and assumepe (7^x0. Let (^i(/?),0) be the local coordinates of p. (4.20) has the unique solution 'zj{s) = zf{p,s), t = s^ under the initial conditions

zno)=cr{p),

a = i,...,n,

t(0) = 0.

This solution gives a smooth curve on Mj passing through p: yp'-t^ypit)

= (zj(p,t),t),

tel.

Thus we obtain a family of smooth curves {jplpeM} on Mj. By the uniqueness theorem of the solution of simultaneous ordinary differential equations, for each point (zy, t) e Mj, there exists just one curve which passes through it. Consequently the map ^:{p,t)^(zj,t)

is a diffeomorphism of M x / onto Mj.

=

izj{p,t),t)

197

§4.2. Infinitesimal Deformation

Since clearly O maps M x t onto M^, in order to see that : M x r ^ M^ is biholomorphic, it suffices to show that Zj{p, t) is holomorphib in p. For this we write Zj{p, t) as a function of t and the local complex coordinates ^J = ^j{p), A = 1 , . . . , n, of /? in the form

Since t = s,by (4.20) we have

~zfUi, t) = -e;(zj(^, f),..., z;(4 0,0. By differentiating both sides of this equation in f f, since ^/(Zj, 0 is holomorphic in z j , . . . , z", we obtain

ddzfu,t)_ dt

d^f

^ sei ,%dzf^''^^'''^''^'

dzfu, t) a^7 *

Put (of(t)=dzfUi, t)/dCt Then (oj'(t) is a solution of the simultaneous ordinary linear differential equations

dt

of(t) = - Z -^{zjiL ^ = 1 dZj

t\ t)cof(t),

a = 1 , . . . , n.

198

4. Infinitesimal Deformation

Since zfU^, 0) = ^r, we have (of(0)=d^T/d[f = 0, Hence the solution co;'{t) satisfies the initial condition: cof{0) = 0. Consequently by the uniqueness of the solution, co]'{t) = 0 identically. Thus dzJUi, t)/d[f = 0, and zJ'ip, t) is holomorphic in p. Therefore ^ : M x ^ - > M f is biholomorphic, which proves that {Mj, /, m) is trivial. I The above proof also applies to the case feB is an arbitrary point of B instead of 0. Thus we obtain the following theorem. Theorem 4.2. Let Mf = m~^ (t). If dim H\Mt, St) is independent of t, and 6(t) = dMt/dt = 0 identically, then (Ji, B, m) is locally trivial Thus under the condition that d i m / / ^ M , ^r) is a constant, we have proved that if ^(r) = 0 identically, the complex structure of M^ does not vary with t. Accordingly we may consider 6{t) as the derivative of the complex structure of M^ with respect to t. In the above argument we have assumed that B is an open interval in IR. But the notion of the infinitesimal deformation is easily extended to the case B is an arbitrary domain in [R". Suppose given a differentiable family {M, B,m) of compact complex manifolds where 5 is a domain of R". Let {xi,... ,Xj,...}, Xj = (Zj, t) = {z],..., z", ^ 1 , . . . , r^) be a system of local coordinates on M, and % the domain of x^. Suppose that on % n^k^0, we have 2'7 =ffkizk, t) =f^k(z],...,

zJ, ^ 1 , . . . , r^),

a = 1 , . . . , n.

As usual, identifying a point of Jt with its local coordinates, we may consider that % = Xj{%) c: C"" X B, and that M = \Jj % is obtained by glueing the domains ^ i , . . . , ^fc,... of C" x J5 by identifying (zy, t) and (z^, t) if Zj = fjk(^k,t).Ut = {Ujt\ Ujt = %nMt9^0} form a finite open covering of M^ = vj-\t),

The tangent space T ; ( B ) of J5 at t consists of all tangent vectors d/dt = i r ^ i c^a/a^A, C;,eU. For d/dteTt(B), put ^jkit)= L

7^,

a=l

Z},=fj,j(zj,t).

(4.21)

dZj

Ot

Then as in the case m = 1, we have {6jj^{t)}eZ\Vit, St). The cohomology class 6(t)eH\Mt,&t) of the 1-cocycle {^^^(0) is called the infinitesimal deformation ofMt along d/dt, and is denoted by dMt/dt: dMt/dt = 6{t). The infinitesimal deformation dMt/dt is independent of the choice of systems of local coordinates. We define the map pt by ^

(d\

Pt is an R-linear map of Tt{B) into H\Mt,

dMt

©J.

§4.2. Infinitesimal Deformation

199

lfpt{d/dt) = OfoTSinyd/dte Tt{B), pt is called the 0-map, and is denoted by pt = 0. If {M, B, vj) is trivial, we see immediately as in the case m = 1 that Pt = 0 identically, using the fact that 9{t) is independent of the choice of systems of local coordinates. Theorem 4.3. If dim H\Mt, 0^) is independent of t, and Pt = 0 identically, then (M, B,VT) is locally trivial Proof Consider the differentiable family Mi = m-\I)

= {J

UjXl

given in (4.8). For simplicity, we assume that / = {(/i,..., ^^)||^i| < r,..., \tm\< f}- We proceed by induction on m as in the proof of Theorem 2.4. Let I"^-' = { ( ^ 1 , . . . , r^) I kil < r , . . . , \t^_,\ < r}, Im =

and

{tm\~r
Then / = / ' " " ' x / ^ . Considering J ' " " ' = /"'"'xOc=/, we see that (m~\l"^~^), I'^~\ m) is a differentiable family. Since Theorem 4.3 is true for m = 1, we may assume that TZT~^(/'"~^) is trivial. Define the map of t!T~\l"'~^)xI^ onto / by (p, tm)^{ru{p), t^) where/7G tz7~^(/'""^), which makes m~\l'^~^)xl^ a differentiable family on the parameter space /. Since by hypothesis m~^{I'^~^) is equivalent to Mx/'^~^ where M = trr"^(0), tz7~^(/'^~^) x / ^ is equivalent to M x / . Therefore in order to prove the triviality ofJ^j, it suffices to verify that ^ / is equivalent to tzT~^(/'"~^) x / ^ . Put a=\

Ot^

dZj

Then the cohomology class ^ ( 0 = Pt{d/dt^) of the 1-cocycle {Ojk(t)} is equal to 0 by the assumption. Consequently as in the case /tz — 1, by (4.14) there exists a 0-cochain {6>,(0}e C^(U„ 6,) such that {Ojk(t)} = slOjit)}. Since Theorein 4.1 remains true also for m ^ 2 (see §7.2(b)), we may assume that the coefficients Of{Zj, ^ i , . . . , r^) in ej{t)= i

e^izj,t„...,tj^

(4.23) aZj

a =l

are C°° functions of z j , . . . , zj, f i , . . . , ^^ in Uj x /. Therefore as in the case m = 1, we obtain a C°° vector field v on Mj such that on each UjXl a Jij v =- i a=l

e^izj, ^ 1 , . . . , ^ m ) - ^ + - f . OZj

dt^

(4.24)

200

4. Infinitesimal Deformation

Consider the corresponding simultaneous ordinary differential equations

"f=-";(','....

' ? ^m/5

ds

ds

= 0,

a = l,...,n,

A = l,...,m-1,

(4.25)

= 1.

Take any point pem-\l'^-') = \J,UiXl'^-'xO, and let (^.(/7), ^ i , . . . , t^_i, 0) be its local coordinates where (iti,..., r^_i, 0) = VT{P). Then (4.25) has the unique solution

^zj{s) = zj{p,s) ^A(^) = ^A,

a = l,...,n, A = l,...,m-1,

under the initial conditions

r^r(o) = fr(p),

a = i,...,n,

\t^{Q) = t^^

A = l,...,m-1,

[^m(0)=0. Thus we get a diffeomorphism ^ofti7"^(/'"~^)x/^ onto Mi: ^'- ( A ^m) ^ {Zj{p, tm), ^1, . . . , ^m-l, ^m).

If, using the local coordinates (^J,..., ^f, t i , . . . , r^_i, 0), we write ^j ( P ? ^m) ~ ^j ( b i? • • • J b i 5 ^1? • • • 5 ^ m - l 5 ^m)?

then since Of(Zj, ti,.,., t^) is holomorphic in z j , . . . , z", we see as in the case m = 1 that z"U],..., ^f, ^ j , . . . , t^) is holomorphic in ^ / , . . . , f f- • Theorem 4.4. ([19]). For a differentiahle family (M, B, m), dim ff H M , €)r) 15 an upper-semicontinuous function oft where Mt = m~^{t).

§4.2. Infinitesimal Deformation

201

By the upper-semicontinuity, we mean that for each s e B, dim H\M„

S,)^dim

H\M,,e,)

if

\t-s\<8

provided that s is sufficiently small. We give proof of this theorem in §7.2(b). From this theorem and Theorem 4.3, the Frolicher-Nijenhuis theorem ([7]) follows. Theorem 4.5. Let {M, B, m) be a differentiable family of compact complex manifolds where B is a domain of W^, and OeB. If H\MO,SO) = 0, with Mo= VT~\0), then for a sufficiently small open interval I with Oel'^B, {MI, I,m) is trivial A compact complex manifold M is called rigid if for any differentiable family {M, B, m) with OeB and txr~^(0) = M, there is an open interval / with Oe I a B such that {Mj, /, m) is trivial. If (Jti, I, m) is trivial, M^ = nT~^{t) has the same complex structure with M = Mo= ti7~^(0) for tel. Thus if M is rigid, the complex structure of M = MQ is invariant under small perturbation of t. By Theorem 4.5, a compact complex manifold is rigid ifH\M, 6 ) = 0, where S is the sheaf of germs of holomorphic vector fields on M. (c) The Case of Complex Analytic Families Let (Jt, B,m) be Si complex analytic family of compact complex manifolds (Definition 2.8) where 5 is a domain of C", and {(zj, t)} its system of local coordinates. Then each (Zj, t) is a local complex coordinate system of the complex manifold M, and

^; =fU^i....,

z;;, ^1,..., tm),

a = 1,..., n,

are holomorphic functions in z ^ , . . . , z^, ^ i , . . . , t^. As in the case of differentiable families, for a tangent vector d/dt = X^^i C),{d/dt^), c^ e C, of B, we put

ejM = i a-1

'-^^^

-^,

Ot

z, =A,(z,, t).

(4.26)

dZj

The cohomology class d(t)e H\Mt,St) of the 1-cocycle {Ojk{t)} is called the infinitesimal deformation along d/dt and is denoted by dMt/dt. The infinitesimal deformation is independent of the choice of systems of local coordinates. d

/ d\

dMt

''-7t^'\7t)=~jr is a C-linear map of T^B) into H^Mf,

S^).

202

4. Infinitesimal Deformation

A complex analytic family (Jt^ B, wr) is called trivial if it is biholomorphically equivalent to (M xB, B, TT) with M = m'^it^) where t^ is some point of B (see p. 61). Similarly we define the local triviality of (M, J5, m). As in the case of differentiable families, t = 0 identically if {M, B, m) is locally trivial. By considering Bc:C"=(R^" as a domain of R^", a complex analytic family {M, B, vr) may be regardedas a differentiable family. The tangent space of B at f considered as a subdomain of K^" which consists of all tangent vectors with complex coefficients is Tt{B)®Tt(B). Since fjici^k, ^1, • • •, tm) are also holomorphic in ^ i , . . . , t^, Sffkiz^, t)/dt = 0 for d/dt = Y.^ Cx(a/a4)e Tt(B), hence pt(d/dt) = 0. Thus the map pt for (M, B, m) considered as a differentiable family coincides with that for (M, B,m) considered as a complex analytic family. For a complex analytic family (Jt, B, xzr) of compact complex manifolds, we have the following theorem similar to Theorem 4.3. Theorem 4.6. If dim H\Mt, 6^) is independent ofteB, and p, = 0 identically, then the complex analytic family {Ji, B, m) is locally trivial. Proof. Proof is similar to that of Theorem 4.3. Let OeB, and take a sufficiently small polydisk A with OeAaB. Put J/^ = tD-"^(A). Then as in (4.8), we have ^A=U ^xA,

(4.27)

7=1

where each Uj is a polydisk independent of t, and (z^, ^ e L ^ x A and (Zfc, t)e Uj^xA are the same point on Jt^ if zj" =f]l(zj^, t), a = 1 , . . . , n. fjki^k, t) =/jfc(zfc,..., Zfc, f i , . . . , trn) is a holomorphic function in the complex variables z ^ , . . . , zJJ, ^ i , . . . , t^. We may assume that A = { ( r i , . . . , r^)EC" ||^i| < r , . . . , | t ^ | < r}. We proceed by induction on m. For this we put A"-^ = {(?!, ...,t^)\\t,\
,\t^_,\ < r},

and

Am={tm\\tm\
We consider A'""* = A'"'^ x O c A'""^ x A^ = A. It suffices to prove that the complex analytic family t3j"\A'^~^) xA^ is biholomorphically equivalent to M^. Put ^ . ,

" dffkJZk, t,,. . . , tj

f^jkKt)-

L

—

_d_ .u-

Since the cohomology class Pt{d/dt^) of the 1-cocycle {Ojk{t)} is equal to 0, there exists a 0-cochain {(9,(0}e C^tt,, 6,) such that {Ojkit)} = 8{ej{t)} where we put U^ = { L^ x r}. As was mentioned in the proof of Theorem 4.3

§4.2. Infinitesimal Deformation

203

above, we may assume that each coefficient O^'izj, ti,..., Oj(t)=

I

t^) in

e-{zj,t,,...,tj^ OZj

« = 1

is a C°° function of z j , . . . , zj, ^ i , . . . , t^. In order to show that tEr~^(A'"~^) x A^ is biholomorphically equivalent to Jt^, it suffices to verify that we can choose a 0-cochain {dj{t)} such that each coefficient ^"(zj, ^ i , . . . , ^m) of Oj(t) is holomorphic in z j , . . . , z", ^ i , . . . , f^. In fact, if each Of(Zj, t^,..., t^) is holomorphic, the map O: tiT~^(A'"~^) x A ^ ^ M ^ defined as in (4.25) by virtue of the solution of the simultaneous ordinary differential equations

ds dtrr

I ds

A= l,...,m-1,

= 0,

= 1,

is biholomorphic and O maps TtT ^(A'" ^) = w- ^(A'" ^) xO identically onto m~\A'"-')^M^. If we write ejM=

I

Of,(zj,t) — ,

a =l

OZj

then Ofki^p t) = (d/dtrn)ffk(zk, 0 with z^ =fkj(zj, t) are holomorphic functions of z],..., zj, ^ 1 , . . . , tm- The equality djk(t) = 6k(t) - Oj(t) is written in the explicit form as

(x = \

dZj

^ =1

dZk

a=l

oZj

namely, " (9z" )8 = 1 OZk

Since dzJ/dzl = df%{zk, t)/dz^, z^ =fkM, t), and Ofkizj, t) are holomorphic in ^ 1 , . . . , ^^, differentiating both sides of the above equality in f^, we obtain

J dzfde'kizk,t)_def(zj,t) f3 = ldZ^

dt;,

dtx

204

4. Infinitesimal Deformation

namely, p^\

dfx

dZj^

a^l

dtx

dzj

Thus putting V\it) = la(^^fi^P O/^h) ' d/dzf on each UjXtaMt, we obtain a holomorphic vector field r};,(t)e H^{Mf, @^) on M^. As was mentioned above, if we consider (M, B, m) as a differentiable family, Pr = 0 also holds identically. Consequently by Theorem 4.3, (Jt, B,m) is locally trivial as differentiable family. Therefore each M, = wr~\t) with r G B is biholomorphicto M = txr'^O). Hence dim H^(Mt, 6,) = dim i/°(M, 0 ) . Note that we have //^(M, 6 ) = 0 for many examples of compact complex manifolds. Such M as dim H^(M, 0 ) ^ 1 is rather exceptional. (1°) The case dimH\M,e) = 0. Then / / ^ ( M „ e , ) = 0, too. Hence 77A(0 = 0 identically. Therefore dOf{zj, t)/dt), = 0 for A = 1 , . . . , m. Consequently O'^izj, t) are holomorphic in z j , . . . , zj, ^ i , . . . , t^. (2°) The case dim H^(M, O) ^ 1. We use the following lemma. Lemma 4.1. For a complex analytic family (Jt, B, m), suppose that dim H^(Mt, (dt) = d is independent oft. Then for a sufficiently small polydisk A with Oe A<=^ B,we can choose a basis {(pi(t),..., <^d (01 of each linear space H\M,, 6,) with a=l

oZj

such that (Pqj(Zj, t) are holomorphic functions of z],...,

zj, ^ j , . . . , t^.

We give proof of this lemma in §7.2(b). Put d

V^{t)= Z c^x(t)
The coefficients C^ACO are C°° functions of ^ i , . . . , t^. We denote d with respect to the variables ti,..., t^ by df. Then on each Uj x A, we have d

d.ef{zj,t)=

m

I
Since dfCPqjiZj, r) = 0 by the above lemma, we have d

_

"^

o = d,d,ef{zj,t)= z
A=l

(4.28)

§4.2. Infinitesimal Deformation

205

Hence dt 1^=1 V ( ^ ) ^^A = 0 . Since Y.7=\ ^^ACO ^^A is a (0, l)-form on the polydisk A, by Dolbeault's lemma (Theorem 3.3), there exists a C°° function Cq{t) on A, for each q such that dtc,{t)= Z «-qAV c^x{t)dh.

(4.29)

A= l

Put ej{t)=^dj{t)-Ht),

. ^ ( 0 = 1 c,(r)
For each r e A, (/^(r) is a holomorphic vector field on M,, hence, 6>y(0 is a holomorphic vector field on UjXtcz M,. Clearly we have Ojk(t) =

Oj(t)-ek(t).

Writing 61(0 = l L i ^r(^> 0^/(5z;, we have ^7(z,, r) = ^;(z„ 0 - I c,(0
Therefore by (4.28) and (4.29), M 7 ( ^ > 0 = 0. Consequently ^/(z,, 0 are holomorphic functions of z j , . . . , z", r^,..., r^. I Corollary. A complex analytic family (Jt, B, m) is trivial if it is trivial as a differentiahle family. Proofs of Theorems 4.1, 4.4 and Lemma 4.1 given in §7.2(b) are based on the theory of harmonic differential forms. Therefore the above theory of deformations of compact complex manifolds is also based on the theory of harmonic differential forms, hence cannot apply to the deformations of complex spaces. Grauert extended the above results to the case of complex spaces by developing the theory of deformations based on the theory of coherent analytic sheaves ([8]). (d) Change of the Parameter Suppose given a complex analytic family { M J M ^ = tz7~^(0, teB} = (Jt, B,VT) of compact complex manifolds, where B is a domain of C". Let D be a domain of C\ and h: s^ t = h(s), se D, a, holomorphic map of D into B. Then by changing the parameter from t to s, we obtain a complex analytic family {Mh(s)\seD} on the parameter space D as follows.

206

4. Infinitesimal Deformation

We denote a point of M by /?, and define the holomorphic map 11 of MxDio BxDhy n:{p,s)-^{t,s)

=

{m{p),s).

Then (JtxD, BxD, U) is a complex analytic family with the parameter space B X D. We have 11"^^, 5) = M, xs. The graph of h G=

{{h{s),s)eBxD\seD}

is a submanifold of BxD, hence a complex manifold. Since the projection P: BxD^ D maps G biholomofphically onto D, we may identify G with D via P. ^ = n~^(G) is a submanifold of the complex manifold JtxD, If we denote the restriction of 11 to ^ again by 11, then (J{, G, U) is a complex analytic family over the parameter space G. Identifying G with D as mentioned above, we obtain the complex analytic family (Jf, D, rr) where 7r = P o n . Definition 4.5. The complex analytic family (^, D, TT) thus obtained is called the complex analytic family induced from (Jt, B, vj) by the holomorphic map h'.D^B. Since Il~\h(s),

s) = Mh(^s)^s, we may consider 7T-\S) = U-\h(s),

S) = M^(,) X S = M;,(,).

Thus {Mft(5)\se D} form a complex analytic family (JV, D , TT). To investigate the relation of the infinitesimal deformation of Mh(s) and that of Mt, assume that OeB,OeD, and h(0) = 0. Taking a sufficiently small coordinate polydisk A with 0 e A c= J5, we represent M^ = IJT~\A) in the form of (4.27) as ^A=U ^xA, where (z,, t)e L^ x A and (z^, t)e L4 x A are the same point of Jt^ if Zj = fjk(zk,t)' Take a polydisk E with OeEc^ D, and put ^ £ = 7r-^(£) = U-\GE)

where G^ = {(h(s),

Jfs is a submanifold of J^A X E. We have J^AX£ = U

^XAXJB,

s)\seE}.

§4.2. Infinitesimal Deformation

207

where (Zj, t, s)e UjX^xE and (z^, t, s)e U^x^xE M^xE if Zj =fjk{Zk, t). Consequently

are the same point on

where {zj, h(s), s)e UjXGE and (z^, h(s), s)e U^xGE are the same point on ^E if Zj=fjk(zk, h{s)). Identifying GE with E = P(GE) via P, we may consider JfE==^~\E)

= [J

UjXE.

Thus J^E is a complex manifold obtained by glueing UiXE,..., UiXE by identifying (z^, s) eU^xE and (z^, 5) e L7y x E if Zj =fjk(Zk-> his)). Then the complex analytic family (J^E, E, TT) induced from (Ji^, A, ttj) by the map /i: JB ^ A is a complex analytic family obtained by substituting t = h(s) into the functions ffki^k, 0 defining Ji^. Theorem 4.7. For any tangent vector d/ds e Ts(D), the infinitesimal deformation of Mh(s) along d/ds is given by ^M,(,) - dt.dM, — = I — T—, \ = i ds dtx ds

( ^ 1 , . . . , ^m) = Hs).

. (4.30)

Proof We put .

, , , _ V ^f%i^k.h,....tm)

^KjkKt)-

L

d

~

ot = \

—^.

Otx

dZj

" dffk{zk.his)) Vjk(s)= I a=l

ds

d —. dZj

dMt/dt), is the cohomology class of the 1-cocycle {d^jkit)), and dMh(s)/ds is that of {rjjkis)}. Since h(s) = (t^,..., t^), we have dffki^k, h{s)) _ - dt;, ds

A=:i ^^

Hence we see that (4.30) holds.

dffk{Zk,tu.--,tm) <5rA

I

(4.30) is just the formula of the derivative of a composite function. A similar formula holds for differentiable families.

4. Infinitesimal Deformation

208

Our Theorems 4.2, 4.3, and 4.6 contain the assumption that dim H\Mt, B^) is independent of t. At first we did not know whether this assumption was essential or not. Since we might expect the local triviality of (M, B, m) in case p^ = 0 if pt(d/dt) = dMJdt is truly the derivative of the complex structure of M,, we suspected that we could get rid of this assumption. But the study of deformations of Hopf surfaces revealed the necessity of this assumption. We will explain this below. Let {M^ C, ID-) be the complex analytic family of Hopf surfaces M^ = 7r~H0 given in Example 2.15. {M, C, m) has the following properties: (1°) If we put U = C -{0}, {Mu, U, m) is trivial; (2°) MQ and M, with r # 0 are not biholomorphically equivalent. Consider the complex analytic family (JV*, C , TT) induced from {M, C, vr) by the holomorphic map s-^ t = s^. Then from (4.30) the infinitesimal deformation ps(d/ds) of 7r~\s) = M^^ is given by dM,2 ds • ( ! ) =

dt dM, ds dt

dM, dt

If 5 7^0, t = s^e U, hence dMJdt = Q because {Mu, U, m) is trivial. Thus we have Ps = 0 identically. On the other hand, (^, C, TT) is not locally trivial. For, 77-~^(0) = MQ and 7r~^{s) = M^ with r = 5^ ?^ 0 are not biholomorphically equivalent. Thus although p^ = 0 identically, (^, C, TT) is not locally trivial. SS^) Since p^ = 0 identically but (^, C, TT) is not locally trivial, dim H\MS^, of computation must not be a constant. By making an explicit dim H\Mt, ©,) ([21]), we have

dimf/HM„e,) = |^'

ty^O.

Chapter 5

Theorem of Existence

Let (M, B,m)hc a, complex analytic family of compact complex manifolds where B is a domain of C. Then the infinitesimal deformation dMt/dt of Mt = n7~\t) is an element of H\Mt, 6^). Consequently, given a compact complex manifold M, if (M, B, m) with O G J 5 C : C is a complex analytic family such that m~\0) = M, {dMJdt)t=oeH\M, 6 ) , where 0 is the sheaf of germs of holomorphic vector fields over M Thus, if there is a complex analytic family {M.B.m) with tJT~^(0) = M, the corresponding element e = (dMt/dt)t=o of H\M, e) is determined. Conversely, given a deH\M,S), does there exist a complex analytic family {M,B,m) with 0 e 5 c: C such that ^-1(0)=M,

(dM\

r ^ ] =0 1 \ dt A=o

This was the next problem in the theory of deformation. In this chapter we explain the development of the theory of deformation in connexion with this problem.

§5.1. Obstructions Lemma 5.1. Let L^ = {(z\ . . . , Z " ) G C " | | Z ^ | < r , . . . , |z"| < r} be a polydisk, and 6 the sheaf of germs of holomorphic vector fields over U. Then H'{U,e)

= 0,

q^h

(5.1)

Proof A germ of a holomorphic vector field i ; e 0 ^ at ZG L^ is written in the form d v = vi—+dZi

d • • + i?„—, (9z„

where the coefficients i;«, 1 ^ a ^ n, are germs of holomorphic functions at z. Thus V can be represented as v = (vu... ,Vn) with v^ e 0^, where we

5. Theorem of Existence

210

denote the sheaf of germs of holomorphic functions over U by €. Therefore

'-\-H^U,0). H''{U,S) = H^U,€)-\-" _—/ .^^ ^ ^ n times

By Dolbeault's theorem (Theorem 3.12), we have

while by Dolbeault's lemma (Theorem 3.3), for any 5-closed (0, ^)-form (pGr(L/,a^^'^"') on a polydisk U, there is a iAGr(U, ^^'^"') such that cp = dil/. Therefore H^(U, 0) = 0, hence H\U, S) = 0. I Let U = { Uj} be a finite open covering of a compact complex manifold M, and suppose that each Uj is a coordinate polydisk. Then by Lemma 5.1 above, we have H\Uj, S) = 0. Therefore, by Theorem 3.5 and 3.6, we obtain the following: Lemma 5.2. H\n,e) = H\M,e),

(5.2)

H\Vi,e)^H\M,@).

(5.3)

Suppose given a compact complex manifold M, and let {Jt, B, m) with 0 G B c: C be a complex analytic family such that tzT~^(0) = M. Take a small disk A with centre 0 such that OG Ac: B, and represent M^= tEr~^(A) in the form (4.27): (5.4)

^ A = U ^-xA,

where each Uj is a polydisk, and (z^, t)e UjXA and (z^, t)e U^xA are the same point on ^ ^ if ^f =fJk{^k-> 0 for a = 1 , . . . , n. Here each fjki^k, 0 = Zfc, r) is a holomorphic function of z j , , . . . , zl, t defined on Uj^ x f^ki^k,..., An^xA7^0. is, by definiThe infinitesimal deformation 6(t) = dAiJdte H\Mt,St) where tion, the cohomology class of the 1-cocycle {6jkit)}eZ\Vit,St) Ut = {UjX t}, and the vector field

a

OZj

is given by

ej,(2j, t)JM5^^ ot

,^ =y^.(

,).

(5.5)

§5.1. Obstructions

211

Note that the functions O^ki^j, t) of Zj,... ,z],t are obtained by differentiating first the functions f%{zj^, t) of zJ,,..., z^, t with respect to t, and then substituting z^ =fkj(^p 0On t/fc X A n L/, X A n ^ X A 7^ 0 , we have the equahties fTkizk, t) =mfjk(zi.,

0 , 0,

a = 1 , . . . , n.

(5.6)

By differentiating both sides of these equahties in t, we have " (9z" eUzi. t) = eUzi, 0 + Z T^Ofj^izj, t),

a = l , . . . , n.

(5.7)

^ = 1 dZJ

As is stated in §4.2(a), these equalities imply that {Ojk(t)} is a 1-cocycle. Next, we consider the equahties obtained by differentiating both sides of (5.7) in t as functions of z],..., z", t. We sometimes write {d/dt)j instead of (d/dt) in order to make expUcit that d/dt denotes the differentiation of a function of z ] , . . . , zj", r with respect to t On U, x A n L/^ x A T^ 0 , we have the following equalities:

_a_^"azf^

dzf _dfl{zj,t)

dzf

dzJ

ph.dzf

dz^'

dzJ

' (5.8)

\dt)j

/^i \ dt )j dzf

\dt/;

\ dt Jj

dt

Consequently

( I Oik{Zi,t)\dt/j

2. p=i

= E p^i

dt

--pOik(Zi,t)^ dZJ

dt

erj{zi,t)—^eik{Zi,t)-\-—-—. dzf

dt

Therefore putting

dt

we have

[£j oUzi, t) = e,j{t) • eUz,, r) + 4(z,, 0.

212

5. Theorem of Existence

Similarly we have

(^)/s<- t) = eyit)-diiz„t)

+07,(2,, t),

and

{7) ^7^ ^U^j, t) \dt/j

13 = 1 dzf (i

dZj

dt

/3 dZj

=1 ef,{zj, t)-^erj{z„ t)+i^^ej,izj, t) j3

dZj

p

dZj

= ej,{t)-erd^„t)+i^^efdzj,t). Accordingly, differentiating (5.7) with respect to t, we obtain the following equalities:

oy ei{z„ t) + ei{z„ t) = e,,{t) • e%{z„ 0+0^(z, t)

Since

we can write the above equalities in the form

erj{z,,t)-erAzi.t)+ i ^^efj,izj,t) (5.9)

= Oijit) 'l^^ef,(zj, t)-ej,{t)' e^{z^ t). /3

^^j

In general we define the bracket of two local holomorphic vector fields t ^ = Z L i < W O and i . = i : ^ , uj{d/dzj) by iv,u-\= i

(v'uf-U'vf)^,

cx = \

dZj

(5.10)

§5.1. Obstructions

213

It can easily be verified that [i;, u] does not depend on the choice of coordinates, [v, u] is bilinear in u and v, and clearly [u,v]=-'[v,u] holds. Putting Oij{t) = Y.l = i ^tji^i, 0 d/dzf and so on, and using the bracket, we rewrite (5.9) as

eij(t)-e,M + ejM = [o,j{t\ejj,{t)i

(5.11)

Substituting r = 0, we obtain ^ f ; ( 0 ) - 4 ( 0 ) + M 0 ) = [^y(0), OjdO)l

(5.12)

Since M = MQ by assumption, identifying Uj with Uj x 0, we may consider 11 = {^} as a finite open covering of M For a given 6e H\M,(d), if 6 = (dMt/dt)t=o, 0 is the cohomology class of the 1-cocycle {Ojk(0)}e Z\Vi,S) above, hence (5.12) imposes a certain restriction on such 6. In order to make clear the meaning of this condition, we put for any 1-cocycle {6i,,}€Z'(U,0)

on Uin Ujn Uk9^0. ^tjk is a holomorphic vector field on UinUjn Uijk} forms a 2-cocycle on U. In fact, first since ^ikj = [ ^ik, ^kj] = ~[ ^ij + ^jk, ^jk] — ~[ Oij, ^jk] = ~ ^i/fc,

^jik = [^jh ^ik] = ~ [ % ^ij + ^jk] = ~ [ % ^jk] =

-Cijk,

^ijk is skew-symmetric in ij, k. Next, on Uh n Ut n Uj n U^ 9^ 0, Cijk ~~ Chjk + Chik ~ Chij = [ ^ij ~ ^h> ^jfc] + [ ^fii, ^ik -' ^ij\

Thus {Cijk) is a 2-cocycle. For arbitrary 1-cocycles {6^^} and {r)jk} G Z ^ ( U , ©), we put

U^.

5. Theorem of Existence

214

Then since Uijk = [ Oij + 7)ij, 6jk + 7]jk] - [ % ^,7c] - [ Vip Vjkl {^ijk} is also a 2-cocycle in Z^{Vi, 0 ) , whose cohomology class ^ is uniquely determined by the cohomology classes 6 of {Oj^} and 17 of {17^^}. In fact if Ojk = ^k- ^p it follows from a simple calculation that

namely, that {^y7c} = 5{[6>y +^fc, 17^^]}. We define the bracket of ^, 17 6

H\M,(d) hy [^,7?] = ^.

(5.13)

[6, 97] is bilinear in 0 and 17, and the equality [0, r]] = —[rj, 6] holds. Putting / = A: in (5.12), we obtain ^^,(0) + Oj^iO) = 0 since 4^(0) = 0. Thus {^jki^)} is a 1-cochain, and (5.12) can be rewritten as HOjkiO)} = U,j,},

Cijk = [OijiO), 0jj,iO)l

Hence we obtain

[e{o),em

= o.

(5.14)

From these results we obtain the following theorem immediately. Theorem 5.1. Suppose given a compact complex manifold M, and 6e H\M, 0 ) . In order that there may exist a complex analytic family {M, B, m) such that trr~^(0) = M, and that {dMt/dt)t=o = 6, it is necessary that [0, 6] = 0 holds. In other words, if [^, 0] 7^ 0, there exists no deformation M^ with MQ = M and {dMt/dt)t=o= 0. In this sense we call [0, 0]eH^{M, 0 ) the obstruction to deformation of M The necessary condition [^, ^] == 0 is obtained by differentiating (5.7) with respect to t and putting t = 0, while (5.7) is obtained by differentiating (5.6) with respect to t. Thus the condition [6, 6] = 0 is obtained from (5.6) by differentiating twice and then putting t = 0. Similarly we obtain infinitely many necessary conditions by differentiating (5.6) m times and then putting Thus we have infinitely many obstructions to deformat = 0 for m = 3, 4, tion of M. In view of this fact we call [6, 0] the primary obstruction. Theorem 5.1 above holds also for a differentiable family, as will be easily seen from its proof.

§5.2. Number of Moduli

215

§5.2. Number of Moduli (a) Wonder of the Number of Parameters Let {M, 5, tu) be a complex analytic family of compact complex manifolds where 5 is a domain of C". Suppose that for any teB, the linear map

of T,{B) to H\M,, e,) is injective. Then since if a/at = 17=1 c^{d/dt^)9^0, dMJdt 7^0, we may consider that the complex structure of M^ actually varies with t. In this case we say that the parameter t is effective. More precisely, Definition 5.1. If for every te B, pt'. T^B) -^ H\Mt, St) is injective, we say that the parameter ^ = ( ^ i , . . . , ^m) of the complex analytic family {Jt, B, m) is effective, and that (Ji, B, wr) is an effectively parametrized complex analytic family. For a given compact complex manifold M, if there is an effectively parametrized complex analytic family {Ji, B, m) where B is a domain of C " containing 0, and t3j"^(0) = M, then since po'. ToiB) -^ H\M, 0 ) is injective, po(^o(^)) is an m-dimensional linear subspace of H^{M, 6 ) . Assume 6 e po{To{B)). Then 0 = po{d/dt) for some d/dt = Z^^j C;,(a/arJ e To{B). Let 5 G C be a variable and put t{s) = (tM,

. . . , tm(s))

= (Ci5, . . . , CmS).

Then if e > 0 is sufficiently small, we have t(s) e B for |5| < s. Consequently, considering the complex analytic family {Mf(5)||5|< E} induced from {Jt, JB, VT) by the holomorphic map s-^ t = t{s), we have M^(o) = Mo = M, and from (4.30),

Therefore by Theorem 5.1, ^ satisfies the equality [6, ^] = 0. Moreover, as was stated above, 6 must satisfy infinitely many conditions. Thus, in general, PO{TQ(B))^H\M,S), and we may expect that m is rather smaller than dim H\M,(d). Since [0, rj] = [r], 6], if 6, v ^ PoiTo(B)), we have

216

5. Theorem of Existence

In this case by we have po{To{B)) = H\M,e). If m = dimH\M,e), H\M,S), Oe any for that considering the above {M,(5)||s|< e}, we see that such OeBciC with {M.B^m) there is a complex analytic family 0. {dMJdt)t=o= m~\0) = M and that In many examples of compact complex manifolds M, the definition of M itself contains several effective complex parameters, by use of which we can construct complex analytic family of deformations of M. For several simple cases, we compared the number m of parameters of M with dim H\M, ©) ([21], Chapter VI, §14). At first, however, since we had no precise definition of the effectivity of the parameter, we used as m the number of the remaining parameters appeared in the definition of M after excluding the obviously ineffective ones. (i) Projective space P". The definition of P" contains no variable parameters. Hence m = 0 . On the other hand, it is well known ([1]], see also p. 244) that f/^P", 0 ) = 0. Thus we have

dimH\P\e).

m=

(ii) Complex tori. Let M = C"/G be a complex torus given in Example 2.8 with G = {I^^!i mjcoj \ mj e Z}, where coj = {coj,..., cuj) G C",7 = 1 , . . . , 2n, are linearly independent over R. Among coi,..., a>2„, there are n linearly independent vectors over C. We may assume that con+i,..., co2n are such ones. If we take complex coordinates of C" such that a = l,

a;„^« = ( 0 , . . . , 0 , r , 0 , . . . , 0 ) ,

«,

then the period matrix of M is given in terms of these coordinates by col

c.? •• • co^

(On

2 (On

•

1 0

0

•

0

0

1

•

• • •

^n

0 0

(5.15)

1

Therefore an n-dimensional complex torus M is determined by n^ parameters (of,j,a = l,...,n. Thus m = n^. In this case we get rid of the ineffective parameters through the coordinate change of C". On the other hand, we can easily compute dim H\M, @) using the formula (3.148): H\M, n^(F)) = H^'^(F). In our case since B = €{T(M)), we have f f \ M , 0 ) = H^'^(r(M)).

(5.16)

§5.2. Number of Moduli

217

G is a group of automorphisms of C", and each element co = {a)\ ..., (o'')e G represents the automorphism of C" given by the translation

Vector fields, differential forms, etc., on M = C " / G correspond in a one-toone manner to G-invariant vector fields, differential forms, etc. on C", respectively. Since holomorphic vector fields 5/(5z", a = 1 , . . . , n, are Ginvariant, they represent holomorphic vector fields on M, and

I-

-I

forms a basis of the tangent bundle T{M) over M. Similarly {dz\...,

dz")

forms a basis of f*{M) over M. Consequently any (pe^-\T{M)) T{M, si{f*{M)®T{M)) is represented as

a-1v=l

=

OZ

Since the Hermitian metric Ii" = i dz"'®dz'^ on C" is G-invariant, it gives a Hermitian metric on M, which we write as YJl,(i = iSa^ dz°'®dz^ where ^a)8 — ^ap' The complex Laplace-Beltrami operator D for this metric is given by

• = -a=\I

d^ dz

dz

since g^" = 5«^, and each term of a^f", « A 5 ^ and b^s in (3.128) contains at least one partial derivative of g«^ = d^p or g^" = 5„^, hence, is equal to 0. Writing a tangent vector ve T^(M) in the form f = Z2=i ^"(^Z^^:"), we introduce the fibre coordinates (C\ . . . , ^") of T(M). We define a Hermitian metric on T(M) by X « r r . Then by (3.142) and (3.143), we have b a = b and

•« = • = - ! T ; ! - ^ . a = l OZ

dz

H°-'(r(M)) = {^ei?«-'(T(M))|n„
(5.17)

218

5. Theorem of Existence

while by (5.17) for (p=lclv


n<,(p = 0 means that

a = l OZ dZ

for all (pt If 0(^^ = 0, by (3.130)

\\dcpn' = {ncpt,
I

i cUz''®z^,c-.ec].

a = l v^\

OZ

Consequently dimH^'Hr(M)) = n^ Hence dim H\M,e) have

(5.18)

J

= n^. Thus we

m = dim H\M,0) in this case. (iii) Hypersurfaces in the projective space. Let ^ = (^o, • • •, ^n+i) be the homogeneous coordinates on P""^^ The number (/x + 1) of the monomials ^0° • • • CTI of degree h = ho+ • • • + /i„+i in (n + 2) variables ^o, • • •, ^n+i is given by M+l=(

,

).

(5.19)

We associate with each point t = (to,..., t^) of the /it-dimensional projective space P^ the homogeneous polynomial of degree h given by tU) = to^o +

t,^o~'^i^"'^t^^Li-

In order that the hypersurface t{^) = 0 has a singularity, (^o,.. -, t^) must satisfy certain algebraic conditions. (See p. 42.) Therefore there is an algebraic subset © g P ^ such that the hypersurface M^ given by r(^) = 0 for r ^ © is a submanifold of P"'^^ Of course M^ is an n-dimensional algebraic manifold. We call M^ a non-singular hypersurface of order h in We shall treat the case n = l later. The case h = l, Mf = P", hence, it contains no parameter. In case /i = 2, M^ is a quadric hypersurface, which can be transformed into the normal form: ^l+- • - + ^^+1 = 0 by a change

§5.2. Number of Moduli

219

of homogeneous coordinates. Thus it also contains no parameter. We consider now the cases n^2 and h^3. First we must calculate the number of effective parameters. Put B = P^ - ©. Then {M^ \teB} forms a complex analytic family, but its parameter space B is not effective. In fact, consider a projective transformation of P""^^

y'C^C'=yL

a=^l

7^^^,, det(r«^)7^o.

Letting (7«^)~^ = ('ya^) be the inverse matrix of (7a^), we have ^« = Z^^o y^p^'/B' Therefore any monomial of degree hin^o,..., ^^+1 is represented as a linear form of monomials of degree h in ^Q, ..., Cn+i- Consequently if we let t(^) = tXCl, t-^ t' is Si projective transformation of P^, which we denote by y:

Since y is an automorphism of P " ^ \ yMt and M^ are biholomorphically equivalent. Since ^ e M^ if and only if C' e yMt, yMt is a hypersurface given by r'(^) = 0, namely, yMt = Mt' = Myt.

Let @ be the projective transformation group of P"^^ Then since each element y G @ is determined by the ratio of {n + lf components y^/s, OL,^ = 0, 1 , . . . , M + 1, ofthe matrix (y^^), dim @ = ( n + 2 ) ^ - 1 . For any point 5 6 B, t=yse®s, @5 is a submanifold of dimension (n + 2 ) ^ - 1 , and as long as the complex structure of Mt = yM^ does not vary. Therefore among ^ components of the local coordinates of B, ((n + 2 ) ^ - 1) ones are ineffective parameters for Mt. Thus the number of the effective parameters of M, is given by m = /x-(n + 2 f + 1 . Hence by (5.19) we have m= y

^

y{n

On the other hand, calculating dim H\Mt, case n = 2, /i = 4, we have

+ 2)\

(5.20)

&t), we see that except in the

dim f f ' ( M „ 0 , ) = ( " ^ ^ ^ ' ' ) - ( « + 2)1

(5.21)

220

5. Theorem of Existence

Thus m=

dimH\M,,e,)

except in the case n = 2, h = 4. We shall give the calculation of dim H\Mt, St) in (c) of this section. In case n = 2, /i = 4, that is, in case M^ is a quartic surface in P^, dim H\Mt, St) = 20 (see p. 308). Since by (5.19), m = ( I ) - 4 ^ = 1 9 , m < dim H\Mt, St) in this case. (iv) Quadratic transform of P^. Take P distinct points qi,..., q^ on P^, and let M be the algebraic surface obtained by applying quadratic transformations Qq^ at each q^, for A = 1 , . . . , i^: M=Qq^'"QjP').

(5.22)

We compute the number of effective parameters of this M when qi,..., move on P^. Choose the homogeneous coordinates on P^ such that gi = ( 0 , 0 , l ) ,

^2^(1.0,0),

^3 = (0,1,0),

q^

^4=(l,l,l).

Assume for simplicity that ^5, • • •, ^^ do not lie on the line through ^2 and ^3, and put q\ = iLuL2,

1),

A = 5,..., K

Them M is determined by 2{v-A) parameters ^51, ^52,..., Cv\, Cvi- In case ^ ^ 4 , M contains no parameters. Thus we have fO

v^A

m= "'

.

"^-7

(5.23)

forM. Next we compute dim H^{M, B) using the Riemann-Roch-Hirzebruch theorem. For a while let M be an arbitrary algebraic surface. We call q = dim H^{M, O^) the irregularity of M, Pg = dim H^{M, Cl^) the geometric genus of M, and Pa~Pg~q the arithmetic genus of M. By the RiemannRoch-Hirzebruch theorem ([10]), dim H\M,

e ) - d i m H\M,

©) + dim H\M,

(d)=\{lc\-5c2),

(5.24)

where c^ e H^(M, Z) is the first Chern class of M and C2 is the Euler number of M Ci is an integer. Classical Noether's formula 12(/?, + l) = c?+C2

(5.25)

221

§5.2. Number of Moduli

also follows from the Riemann-Roch-Hirzebruch theorem. By (5.24) and (5.25) we obtain dim H\M,

6 ) = 2c2-U{pa

+1) + dim H\M,

0 ) + dim H\M,

O).

(5.26)

Again let M be the algebraic surface given by (5.22). If M = P ^ H H P ^ 0 ) = 0 as stated in (i). By (3.157) dim H\P\

0 ) = dim H%M,

a\K)).

(5.27)

Let {(zj, Zj)} be a system of local complex coordinates on M. Then ((/e H^{M,Ct\K)) is represented as

^ a = \ /3,y=l

namely, r/^ is a covariant tensor field i/^«^^, \\f^^y = - if/ay^ which is holomorphic on all of M. Consequently H^(P^, 0 ) = 0 follows from the fact that there is no non-zero holomorphic tensor field on P^, which is easily verified as follows. Let ^ = {^ G P^ I ^j 9^ 0}, for J = 0, 1,2. Then each Uj is a copy of C^ and P^= Uo^ UiU U2. Putting z^ = f 1/^0 and z^ = Ci/^o, we get the coordinates (z\z^) on UQ. Similarly ( w \ w^) = (fo/fi, ^2/^1) gives the coordinates on U,, On UonUu 1

1

z^ = w

^,=0

^2 = 0

^0 = 0

Figure 1

(5.28)

5. Theorem of Existence

222

hold. A holomorphic covariant tensor field if/ of, say, rank 2 on P^ is written on (7oas(/f = Xa,^=i ipoap dz"" ® dz^ and on ^ i as (/^ = Xl/3=i ^ic.(3dw'^®dw^, where i/^oa^ and 1/^1^/3 are holomorphic functions on UQ and L/j, respectively. On UonUi, we have -

/

^

/ V y,5 = i

dz

^ dz^

while by (5.28) we have

oZ

aZ

dZ

oZ

Thus each aw^/^z", a, 7 = 1, 2, can be extended to a holomorphic function on L/i, which vanishes on the line ^0 —0. Therefore (/^oa^ is extended to a holomorphic function on IJ^^ (/j = P^-{(0, 0, 1)} which vanishes on ^0 = 0This extended holomorphic function is again extended to all of P^ by Hartog's theorem (Theorem 1.8). Thus we have shown that i/fo«^ is a holomorphic function on all of P^, vanishing on the line ^0 = 0. Since a holomorphic function on a compact complex manifold is a constant, i/^oa^ vanishes identically, hence, a covariant tensor field \\s is identically equal to 0. A holomorphic differential form is a holomorphic skew-symmetric covariant tensor field. Consequently / / ^ ( P ^ ^ ^ ) = 0, and /f^(P^a^) = 0. Thus the irregularity q, the geometric genus Pg and the arithmetic genus Pa of P^ are all equal to 0. Similarly we can determine holomorphic vector fields on P^. Let ve H^{P^,@). V is written as v = vl{d/dz^)-hvl{d/dz^) on UQ, and as i; = vl{d/dw^) + vl{d/dw^), where vS = vS{z\ z^) and v^ = uf ( w \ w^), a = 1, 2, are holomorphic functions on Uo = C^ and Ui = C^, respectively. On L^o*^ (7i, we have « = 1,2.

-T. Vi = ^ o T T + ^ o -dZ dZ

Consequently by (5.29) we have

\w

w/

\w

w/

w/

\w

Put

vkz\z')=

I h,k=0

a,^{z')'{z')'

and

vl{z\z')=

I h,k=0

b,,iz')'(z')\

§5.2. Number of Moduli

223

Then by (5.28)

h,/c=0 (W j

Vi{w,w)=

X

r..Mh^k-i

h,k=0

•

(W )

We must determine a/,fc, 6^^ such that these two power series contain no terms of negative exponent for w\ Thus a holomorphic vector field v is written as

OZ + (C4+ CsZ^ + C6Z^+

CjZ^Z^+Csiz^)^;-^,

where C j , . . . , Cg are arbitrary constants. Therefore we obtain dim H%P^, ©) = 8.

(5.30)

Since the Euler number of P^ is equal to 3, from (5.26) we have H\P^, 6 ) = 0. In order to compute dim H\M, S) for M = Qq^' • • Qq^{P^), it suffices to see how each term of (5.26) changes under a quadratic transformation. For this take an arbitrary point qe M, and put M = Qq(M). As stated in Example 2.12, M is a surface obtained from M by replacing q by the projective line Qq{q) = S = P^:

M=

{M-{q})uS.

Therefore M-S = M-{q}. First we investigate the correspondence between holomorphic tensor fields on M and those on M. Given a holomorphic tensor field ij/ on M, by restricting (^ to M - 5, we get a holomorphic tensor field if/' on M - {q} = lH-S. By Hartog's theorem, ij/' is extended to a holomorphic tensor field ij/ on M which coincides with ip on M-{q} = M - S. Thus to a holomorphic tensor field ij/ on M corresponds a holomorphic tensor field ijj on M. Conversely, given a holomorphic tensor field ip on M and by restricting ij/ to M — {q}, we obtain a holomorphic tensor field il/' on M - S = M - {q}. We examine the possibility of extension of il/' to all of M. Let (zi, Z2) be local coordinates with centre q, and put W - {(zi, Z2) | |zi| < e, |z2| < e) for a sufficiently small e > 0. Then

w'=a(w^)={(z.,z2,n6wxp'i^,z2-f2z,=o},

224

5. Theorem of Existence

where Uuii) denotes the homogeneous coordinates of ^eP\ Putting U, = U\i2^0}^Sind U2 = U\Ci9^0}, we have P'= U,u U2. Therefore putting W^ = Wn WxU^,a = l, 2, we have W=WiU W^. Let Wi = ^1/^2 be the non-homogeneous coordinate of the point t,^U\. Then we have ^ 1 = {(^1, ^2, Wi) G Vl^ X ?7i I Zi = W1Z2}. Thus we may use (wi, Z2) as coordinates on W^. We write W2 instead of Z2. Then W^ = {(>Vi, W2)GC^||W2| < e , |>ViW2|< e}, and (?~^: Wx-^W'v$> given by

5 n 1^1 is a submanifold of Wi defined by W2 = 0: Sn ^ 1 = {(wi, 0)| vvi eC}. Therefore W,-S

= {(wuW2)e Wi\w29^0}'^

W-{q},

where the inclusion map ^ is given by (wi, ^2)-^ (zj, Z2) = (W1W2, W2). Let i/^ be a covariant tensor field of, say, rank 2. Then on W, if/ is written as 2

^=

I

il/ap(zuZ2) dz^®dzf3,

«,/3 = l

where il/^isizu ^2) are holomorphic functions of Zi, Z2. Therefore if we represent (A' on Wy-S<=^ W-{q} as 2

then dW^

7,6 = 1

dWp

while we have dZi dWi

= W2,

dZi dW2

=Wi,

dZ2 dWi

= 0,

dZ2

=1.

dW2

Consequently ifjapi^u ^2) is extended to a holomorphic function on Wi, hence if/' is extended to a holomorphic tensor field on Wi. Since similar result holds for W2, «A' is extended to a holomorphic tensor field ij/ on M.

§5.2. Number of Moduli

225

Thus, if ifj is a holomorphic covariant tensor field, i/^' is extended to a covariant tensor field on M. Consequently there is a one-to-one correspondence between the holomorphic covariant tensor fields \p on M = Qq{M) and those if/ on M. Next consider the case e/^ is a holomorphic vector field on M. In this case ip is represented on W as dZi

dZ2

w h e r e JAIC^I^ ^2), ^li^u 2^2) a r e h o l o m o r p h i c f u n c t i o n s o f Zi, Z2. Since Zj = >Vi>^2, 22 = >^2 a n d >Vi = Zi/z2 o n \ y i - 5 c : V l ^ - { g } , w e h a v e

_d__J_ dZx

d

W2(9Wi'

_d__

Wi d

d

dZ2

W2 dZi

dZ2

Therefore we have ^ =

1-—^4f2{^

W2

1^2,^2)—-'

dWi

dW2

Expanding 1^1(^1, ^2), ^2(^1, ^2) into the power series in Zi, Z2, we see that the coefficient of d/dWx in ip has the form (Ai(0,0)-w,(A2(Q,Q) , . , u- r .• ha holomorphic function W2

on W^i. Consequently i/^' is extended to a holomorphic vector field on W^ if and only if JAI(0, 0) = (^2(0, 0) = 0, that is, i/^ vanishes at q. Thus a holomorphic vector field (/? on M = Qq{M) corresponds in a one-to-one manner to a holomorphic vector field j/^ on M which vanishes at q. H^{M, Cl^) is the vector space of all holomorphic covariant vector fields on M, H^(M, Cl^) is that of all holomorphic skew-symmetric covariant tensor fields of rank 2, and H^{M,Ct\K)) is that of all holomorphic covariant tensor fields ip^xpy of rank 3 which are skew-symmetric in the indices j8, y. Therefore from the above results, as to M = Q^- • • Qq^{P^), we see that

H\M,n'),

H%M,[l^) and H\M,n\K))

are isomorphic to

H\P\a'),

/ / ^ ( P ^ a ^ ) and H^{P^,n\K)) respectively. Consequently the irregularity q, the geometric genus Pg and dim H^{M, S) = dim H^(M, [l\K)) all vanish, hence also the arithmetic genus Pa of M vanishes. Again from the above consideration, H^(M, ©) is isomorphic to the subspace of//^(P^, 0 ) consisting of holomorphic vector fields which vanishes aiq;„X = l,..., v. Therefore by (5.30) we have

dim//«(M,0) = {^ ^^'

226

5. Theorem of Existence

On the other hand, since the Euler number of q^ is equal to 1, and that of QqA^\) = P^ is equal to 2, the Euler number increases by one if we replace a point qx by Qq^iq^). Therefore the Euler number C2 of M = Q^^ • • • Qq^{P^) is equal to 3+ ^. Substituting these values into (5.26), we obtain

6imH\M,®)=\:

„

; ;

Consequently by (5.23) we have m=

dimH\M,(d).

So far for several examples of compact complex manifolds M, we have compared dim H\M, @) with the number of the parameters of M roughly computed. Except only one example—the quartic surfaces in P^—we have seen that the equality m = dim H\M,e)

(5.31)

holds. This seemed very strange since at that time nothing was known concerning the existence of deformation of M. As stated at the head of this section, we had expected that in general the number m might be rather smaller than dim H\M,S). The effort to explain this strange phenomenon by proving the theorem of existence for deformations contributed much to the subsequent development of the theory of deformation. In this sense it may be said that the theory of deformation was at first an experimental science. A complex manifold of dimension 1 is just a Riemann surface. The idea of deformation of a Riemann surface goes back to Riemann. Let M be a compact Riemann surface of genus g. If g = 0, M is P \ and if g = 1, M is a 1-dimensional complex torus. Therefore we assume in the following that g ^ 2. According to Riemann's well-known formula, the complex structure of a Riemann surface of genus g = 2 depends on 3 g - 3 parameters. According to the theory of Teichmiiller ([30]), the number of parameters of a compact Riemann surface M is given by m = dim H%M,

€(T''(M)®T''{M))).

Let {zj} be the system of local complex coordinates, and Uj the domain of zj. Then an arbitrary i/j e H%M, OiT'^iM)® T'^iM))) is written on each Uj in the form if/ = iljj dzj ® dzj,

where ipj is a holomorphic function on Uj. Consequently ijj is called a holomorphic quadratic differential. There exist g linearly independent

§5.2. Number of Moduli

227

Abelian differentials of the first kind on M. Let COQ be one of them, and ! = {(OQ) the divisor of COQ. The degree of ! is given by deg!=2g-2. Since we assume g ^ 2 , we have ! > 0 . Since on each Uj, we can write ^0 = ^oj dZj with holomorphic (OQJ, if we write a holomorphic quadratic differential ij/ as (/^=/- 0)00),

/ is a meromorphic function on M and we have (/) + 2! ^ 0. Thus we have fe ^(2!). Conversely i f / G ^(2!), if/ = / • o;® a; is a holomorphic quadratic differential on M (For notation, see [17]). Therefore H%M, €{T''{M)®T''{M)))

= ^(2!).

By the Riemann-Roch formula, we have dim ^(2!) - d i m ^ ( - ! ) = deg 2! - g + 1 . On the other hand, since ! > 0 , ^ ( - ! ) = 0, and d e g ! = 2 g - 2 , hence dim ^(2!) = 3 g - 3 . Thus we obtain m = dim//^(M,^(r*(M)®r*(M))) = 3g-3. By (3.157) dim H\M,

8 ) = dim H\M,

while K = T'^iM) in our case. Consequently ^\K) hence

n\K)), = €{

T''{M)®T''{M)),

m = dimHHM,e). Thus if M is a compact Riemann surface, dim H\M, 6 ) coincides with the number of parameters of M already known long before. (b) The Number of Moduli In the preceding section, we computed the number of parameters for several examples of compact complex manifolds, without giving their precise definition. In this section we shall give the precise definition. We begin with the definition of the completeness of a complex analytic family.

228

5. Theorem of Existence

Definition 5.2. Let {M, B,m) be a complex analytic family of compact complex manifolds, and f e B. {M, B, m) is called complete at feB if for any complex analytic family {M, D, ir) such that D is a domain of C^ containing 0 and that Tr~^{Q) = m~\t^), there are a sufficiently small domain E with OeEc: D, and a holomorphicmap h: s^ t=' h{s) with h{0) = 0 such that (J^E, E, TT) is the complex analytic family induced from {M, B, m) by h where J^E = "^~\E). Ns = 7T~\s) is a deformation of No= Mt^= m~\t^). If {J^E.E.TT) is induced from {M, B, m) by h, Ns = Mh(^s) — "^^^(5)) for s e E. Consequently if (M, B, VT) is complete at t^ e B, (M, B, xu) contains any deformation Ns of MfO provided that s is sufficiently small In this sense we may say that {Jt, B, m) contains all sufficiently small deformations of Mt(^= xxT~^{t^), Definition 5.3. (Jt, B, m) is called a complete complex analytic family if {M, B, m) is complete at every point te B. A complete complex analytic family {M, B, m) contains all sufficiently small deformations of each Mt = m~^{t) for t e B. Definition 5.4. Let M be a compact complex manifold. If there is an effectively parametrized and complete complex analytic family {M, B,m) with VT~\0) = M where B is a domain of C " containing 0, we call m = dim B the number of moduli of M and denote it by m{M). We don't define the number of moduli of M if there exists no such family. The number m(M) of moduli of M is the number of effective parameters of the complex analytic family (Ji, B, m) with m~\0) = M which contains all sufficiently small deformations of M. Thus the number of parameters of M in the precise sense is considered to be its number of moduli m{M). If the number of moduli of M is not defined, we consider that the number of parameters of M cannot be determined. We will explain the reason of this later (p. 314). We must verify that the number of moduli of M does not depend on the choice of complex analytic family used for the definition. For this, suppose given another effectively parametrized and complete complex analytic family (^, D, TT) where D is a domain of C^ containing 0, and 7T~\0) = M. We must show / = m. Since 7T~\Q) = M = m~\Q), and {M,B,m) is complete, there is a domain E with OeEciD such that (JfE, E, IT) with NE = 7r~\E) is the family induced from {M, B, m) by a holomorphic map h.E^B with /i(0) = 0. Put Mt = m~\t). Then iV, = 'TT~^{s) = Mh^s)' Since by assumption s == ( ^ j , . . . , 5/) is an effective parameter of (#, D, TT), putting t = h{s) for s e E, WQ see that dN -^e

H\Ns, ©J = H\M,, 0,),

M = 1, • • •, /,

(5.32)

§5.2. Number of Moduli

229

are linearly independent. On the other hand by (4.30), we have

hence / ^ m. Interchanging M and ^ , we have also m ^ /. Consequently / = m, and the number of moduli m(M) of M does not depend on the choice of {M, B, m) used for the definition. Moreover since / = m, det(ar;v/<5'Sya) A,/Lt=i,...,m 7^ 0 by (5.32). Therefore putting A = h{E), we see that A is a domain of B and that h.E-^ts. is holomorphic. If we take E sufficiently small, then A is also small, hence M^= ti7~\A) is represented in the form (4.27):

M^ = U^,xA. Assume that (z,, t)e UjXA and (z^, t)e t 4 x A are the same point of ^ A if Zj =fjk{z]^, t). Then, as stated before (p. 207), we have

^E=UUjXE, where {Zj,s)eUjXE and {zj^, s) e Uj^ x E are the same point of J^E if ^j =fjk(zk,h(s)). Considering the biholomorphic map h: s^ t = h{s) as the coordinate transformation of the parameter space, we see that (JV^, E, TT) and (M^, A, m) may be considered as the same complex analytic family. In this sense, if for a given compact complex manifold M, an effectively parametrized and complete complex analytic family (Jt, B,m) with m~\0) = M exists, then (M^, A, tu) is uniquely determined by M provided that A is sufficiently small with 0 G A c: B. For the examples of compact complex manifolds given in the precediiig section, we have seen that the number of effective parameters m of M is equal to dim H\M, (d) except for one example. Since it is difficult to believe that for so many examples the equality m = dim //^(M, 0 ) occurs only by accident, it might be reasonable to expect that the number m of parameters will coincide with the number m(M) of moduli of M. Let ti,..., t^ be effective parameters contained in the definition of M, and put t = (ti,...,tm)' Further let M^ be the manifold corresponding to the parameter t, and B the domain of t Then {Mt\te B} is a complex analytic family, which we denote by (M, B, TT). In order to prove m = m(M), we must prove that {M,B,m) is complete and effectively parametrized. For the completeness of a complex analytic family, we have the following theorem. Let {M^B^m) be a complex analytic family of compact complex manifolds where B is a domain of C", and p^: T^B) -» H\Mt, 0,) the linear

230

5. Theorem of Existence

map defined in (4.22) for Mt = m \t), that is,

Theorem of Completeness ([20]). If pt^ is surjective at feB, p,o(T,o(B)) = H\Mt% @fO), then {M, B, m) is complete at t^.

that is,

We give the proof of this theorem in the next chapter. Using this theorem we see that ii{M,B,xu) is effectively parametrized and m = dim H\Mt, 8^) for every t e B, then we have m = m{M). In fact, if {Jt, B, m) is effectively parametrized, pt is injective, hence dimpt{Tt(B)) = m. Consequently pt{Tt{B)) = H\Mt, St), and by the theorem of completeness, (Jt, B, ur) is complete. Thus in order to prove that m = m{M) for the examples given in the preceding section with m = dim H\M,S), it suffices to verify that (M, B, m) = {Mt \te B} is effectively parametrized. (i) Projective space P". Since H\P'',S) = 0,by the theorem of completeness, the complex analytic family {P"} consisting of only one member P" is complete. Hence m(P") = 0. (ii) Complex tori. Let M be a complex torus of dimension n with the period matrix of the form (5.15): 't\

••

t\ •• 1,0 • • _0

• •

tf t"n

0 1_

Put ^ = (^a)a,)8 = i,...,nj ^nd wHtc M^ for M. Let

be the /cth row of the period matrix. Namely cof{t)

Jtf, Uf_„'

; = l,...,n ; = /t + l,...,2n.

Then Mt = C/Gt where G^ = {Xj"i myCeji^(0|'W;GZ}. In vectors (0j{t),j = 1 , . . . , 2n, are linearly independent over U if and only if det Im ^ T^ 0 where lmt = {lmt^)^^p = ir,. Let B = {r|det Im r > 0 } . Then {Mt\teB} forms a complex analytic family. This is verified as in the case n = l given in Example

§5.2. Number of Moduli

231

2.13. In fact, let ^ be the group of automorphisms of C" x B consisting of all automorphisms defined by (z, o ^ ( ^ + Z

7 = 1,...,2n.

^j^j(0, n ,

Then ^ acts properly discontinuously without fixed points, hence the quotient space J/ = C " x B / ^ i s a complex manifold. The projection (z, 0 ^ ^ of C" x B onto B induces a holomorphic map m of Jt onto B. Then it is clear that (M, B, wr) is a complex analytic family and that m~\t) = C " x r / G , = M,. In order to compute the infinitesimal deformation of M^, we take the following system of coordinates on Jt. Let 11 be the canonical holomorphic map of C" X JB onto M = C" xB/^.Uis a. locally biholomorphic map. Let {^fc} be a locally finite open covering of M where ^^ is assumed to be a sufficiently small domain. Then n~^(^fc) consists of infinitely many mutually disjoint domains ^k^, ^k^,... of C" x R Choose one of them, say, ^fcp and let (zfc,..., Zfc, r) be the restriction of the complex coordinates ( z \ . . . , z", 0 of C x B to ^fc^. Since 11: ^k^-^^k is biholomorphic, we may consider (zfc,..., Zfc, 0 as the complex coordinates on ^k- We use these (z^, t) = (zfc,..., Zfc, 0 as local complex coordinates of M on ^k- On ^^ n ^fc we have

Zi = ^/c + I mikojjit),

m]ke\

namely zf==z^ + m r ^ + I

i8 = l , . . . , n .

mr,r^,

(5.33)

a=\

Put a=l

and define ^ikCO as Oik(t)= z — - ^ — i 7 ? = ^ ' ^ i ; ? y=i

5^^

dz[

(^'^^^

dZi

Then the infinitesimal deformation dMt/dt^ is the cohomology class 6{t)e H\M,,(dt) of the 1-cocycle {(9,fc(0}. Let ^ , = M{T{M,)), "^t is the sheaf of germs of C°° vector fields of the form Za
%,) ^ r(M„ a ^ , ) / 5 r ( M „ ^ , ) .

(5.35)

232

5. Theorem of Existence

Since ^^ is a fine sheaf, from the exact sequence

we obtain the exact sequence of cohomology groups

from which (5.35) is derived. Therefore a ^-closed vector (0, l)-form (p(t) e r ( M „ a ^ , ) corresponds to e{t)eH\M,,^,) if 8''(p{t) = e{t). Let U, = { Ukt I Ukt = ^fc n M, 7^ 0 } . If on each Ukt e U t, we can choose C vector field ifj^eTiUkty'^t) such that

then since dilji = dil/k on each UHD Ukt'^0, we obtain a (p(Oer(M,,^'^f) which coincides with dil/^ on each U^r- Then by the definition of 5*, 8'^(p(t)=e{t). For, if we put c^^fiAJe C'^CU,, ^ , ) , we have 5c^= (^(r) and Sc' = {e,,(t)}. Such a 0-cochain {il/k}^ C^i^h^^t) as il/k-^i = ^ik{t) is obtained easily as follows. By (5.33),

By assumption det(rf - f^)„^^==i... „7^0. Therefore letting (M«)a,^=i,...,n be the inverse matrix of {t^ - r?)a,^=i,...,r„ we have

7=1

7=1

Consequently putting

since a/azf = a/^z^, we obtain from (5.34)

Since dzl = (if^, and dzl = 0, we have

7=1

dZ

§5.2. Number of Moduli

233

It is obvious from (5.18) that (p{t) e T{Mt, 5'^^) is a harmonic vector (0, 1)form: (p{t)en^'\T{Mt)). Thus via the isomorphism (5.16):

//HM,e,)^H^'HT(M)), the infinitesimal deformation pt{d/dt^) = dMt/dt^ e H\Mt, St) corresponds to the harmonic vector (0, l)-form Y.l=i " " dz^ d/dz^. Since det(M")7^0, Pt{d/dt^), a, ^ = 1 , . . . , n, are linearly independent. Therefore n^ parameters ^«, a, j8 = 1 , . . . , n, are effective, and miMt) = m = dim H^M^, 0,). However the situation for n ^ 2 is somewhat different from that forn ^ 1. In case n = 1, M^, with Im ^ > 0, is an elliptic curve, and, as stated in Example 2.14, in the complex analytic family {M^ 11 e ^"^}, Mt' and M^ are biholomorphic if and only if t' =

at + b -, ct-\- d

a,T),c,deZ

and

ad-bc=l.

Let r be the group of all linear transformations of the above form. Then F is properly discontinuous, and ¥t/T = C Thus up to the biholomorphic equivalence, an elliptic curve corresponds in a one-to-one manner to a point T G C = IHl"^/r. We express this fact by staying that C is the space of moduli of elliptic curves. The number of moduli m(M^) = 1 is the dimension of the space of moduli C. Consider next the case n ^ 2 . Let Z" be the set of all n x n integral matrices a = (a^), a ^ e Z . Then in the above notation, M^ and M^ are biholomorphic if and only if t' = {at+b){ct + dy\

a,b,c,deZr,,

det(

(c

. ) = 1- (5-36)

. ' ) = '•

Let r be the group of automorphisms y: t-^ t' of the form (5.36). Then for n ^ 2, the group F of automorphisms of B is not properly discontinuous ([28]). Moreover in any neighbourhood U of any point seB with se Uc: B, there is a te U such that TtnU is an infinite set. Consequently, in contrast to the case n = l, B/T is not even a Hausdorff space. Thus we cannot define B/T as the space of moduli. Let yj, p = l,2,... be infinitely many points of F^ n U. Then M^^^, M^^?,... are biholomorphic to one another. Thus/or any small neighbourhood U, there are infinitely many points y^t, 72^ • • • ofU such thatMy^t, My^f> • • • have the same complex structure ([21]). Nevertheless B is an effective parameter space.

234

5. Theorem of Existence

In view of this example, we suspected that there would exist a complex analytic family which is not locally trivial, but all members M^ = ^~\t) of which are biholomorphic to one another. But, as already mentioned, Fischer and Grauert proved that this cannot happen. (c) Complex Analytic Family of Hypersurfaces In order to prove that for a non-singular hypersurface M^ of order h in P""^^ with n ^ 2, /i ^ 3, the number m{Mt) of moduli is defined and coincides with m given in (5.20) except for the case « = 2,/i = 4, we need several results from the general theory on complex analytic families of non-singular hypersurfaces. First we explain them. In general let 5 be a submanifold of a complex manifold W. By the codimension of S, we mean the difference of the dimensions of W and 5. Then a non-singular hypersurface of order /i in P""*"Ms a submanifold of P"^^ of codimension 1. Let W be a given compact complex manifold of dimension n + 1 . A complex analytic family {Jt, B, m) of compact complex manifolds is called a complex analytic family of hypersurfaces in W if it satisfies the following conditions: (i) (ii)

Each Mt = m'^t) is a submanifold of codimension 1 of \y for r G B. There is a holomorphic map (^ of M into W whose restriction to Mt coincides with the inclusion Mt "=-> W.

Since we are only interested in the infinitesimal deformations, it suffices to take a sufficiently small polydisk Ac: B, and consider {M^, A, m) instead of {Jt, B, m) where we put J^A = tix'^A). Moreover we may assume that Ji^ has the form /

where (zj, t)e UjXA. and (z^, t)e L4 x A are the same point of M^ if zf = fjki^k, 0- Let {Wi,..., W;,...} be a finite open covering of W, where Wj is a coordinate neighbourhood, and let Wj = (w^^,..., w") be the local coordinates defined on Wj. Taking Uj and A sufficiently small, we may assume that <|)(^xA)c: WJ for7 = l , . . . , / , and that ^(M^)n[Wj] = 0 f o r j ^ Z + l . O is represented on each L^ x A as CD: (zj, 0 ^ «

. . . , wJ) = (
Since O induces the inclusion M ^ ^ W, rank—-1 d{Zj,...,Zj)

3rr=«-


§5.2. Number of Moduli

235

We denote a point of M by p, and define the map cp of M into WxB

^:p^^{p)

by

= (^{p),7T{p)).

Then O is represented in terms of local coordinates as {zj, 0 ^ ( w ° , . . . , w;, t) = {
cpjizj, t), t).

Therefore we have ^ d{w]...,wj,t,,...,tj ^ rank ——^

^ .. ^ =m + n = dim M.

d{Zj, . . . , Zj , ti, . . . , tm)

Consequently O is a biholomorphic map of M^ into \y x A, and, identifying M^ with 0 ( J^A), we may consider that J^A is a submanifold of codimension 1 of WxA. Then

is considered as the graph of VT~^: t-^ Mt. If we take WJ and A sufficiently small, M^ is given on each W) x A by a single minimal equation Sj(wj,t)

= 0.

Here 5,(w^, r) is understood to be a non-vanishing holomorphic function if Wj x^nM^ = 0. For a fixed ^, 5,(w^, 0 = 0 gives a minimal equation of Mt in W;. 5,(w;, t) = Sj(w^,..., wJ, t) is a holomorphic function of Wj,... ,Wj, ti,..., tm, and at least one of the partial derivatives dSj{Wj, t)/dwf, A = 1 , . . . , n, does not vanish for each Wj e M^ On \ ^ x A n \yfc X A 7^ 0 , we have SjiWj, t) = Fj^(w, t) • S^(wk, 0,

(5.37)

where i^7c(w, t) is a non-vanishing holomorphic function defined on (WJD Wfc) XA, and Wy, w^ are local coordinates of we WJD W^. It is clear that F,fc(w, 0 = Fijiw, t) • Fjk{w, t)

for

w G Wi- n WJ- n W^.

(5.38)

Thus for every ^ G A, we can define a complex line bundle &^t over W with transition functions Fjj^{w, t). ^t is a complex line bundle over W defined by the divisor M^: ^t = [M^], and S{t) = {Sj(Wj, t)} is a holomorphic section of ^ , over W.

236

5. Theorem of Existence

Fix r G A for a while. Since M^ = U^. Uj x t, if we put UjXt= Up, Up is a coordinate polydisk on M^. We denote the point ZjXte UjXt= Up simply by Zj, and consider z, = (z],..., zj) as the local coordinates on M^. If we put (Pj(Zp t) = (cp%Zj, 0 , . . . ,
the inclusion : M^ ^ W is represented on each Up c: Mf as ^:zj-^wj

= (pjizpt).

Put F,fc(z) = Fjj^i^iz), t),

z e Up n U^,

(5.39)

Then Ftjkiz) is a non-vanishing holomorphic function on Up n L/fcr, and from (5.38) we obtain immediately z e Uit n Up n 17^,.

Ftikiz) = Ftij(z) • Ptjkiz),

Therefore we can define a complex line bundle F^ on M^ with transition functions Ftju{z). Ft is the restriction of ^t to M^. In general when t moves, the complex structure of M^ as well as the position of M, in W varies. The change of the position of M^ is called a displacement. On each neighbourhood Wp Mt is given by the equation Sj(wp t) = 0. In order to define the infinitesimal displacement along d/dt for an arbitrary d/dt e Tf(A), consider the partial derivative dSj(wp t)/dt, and put

. / , , , ) - ( ^ ^ ) \

01

(5.40) / wj =

,pj(zj,t)

Applying d/dt to both sides of (5.37), we have =-Fji,iw, t)

—^ ot

+ ^^— dt

Skiw^, t).

at

Restricting these equalities to Mj we obtain (TjiZp t) =

Ftjk{z)(Tk{Zj,,t),

since 5fc(wfc, 0 = 0 on M,. This means that (T{t) = {o-j{zp t)} is a holomorphic section of F, over M^. Definition 5.5. (r(t)£H^{Mt, ^(Ft)) is called the infinitesimal displacement of the submanifold M, of codimension 1 of W, and is denoted by (aM,c

w)/dt.

§5.2. Number of Moduli

237

We denote by pd,t the linear map d/dt-^a(t) namely,

of r,(A) to /f^(M„ 0(Ft)),

Next we will consider the relation between the infinitesimal deformation and the infinitesimal displacement. Let T{W)t be the restriction of the tangent bundle T(W) of W to M^. T{W)t is a holomorphic vector bundle over Mf. Since the tangent space T^iMf) at zeM^ consists of all tangent vectors of W^ at z which are tangent to M,, the tangent bundle T{M^) = U^eM, TAM,) of M, is a subhundle of T ( \ ^ ) , = U.eM. T^W). A local holomorphic section ^ of T{ W)t is represented in terms of local coordinates as A=o

aW^-

where the coefficients v^{z) are local holomorphic functions on M,. Define Tj{Zj, t) for this ^ by

A=0

\

dWj

J ^. = ^.^^.^t>^

Then rj(Zj, t) is a holomorphic function on Uy, c: M^, and from (5.37) we obtain Tj(Zj,t)

= Ftjk{z)Tk{Zk,

t)

on ITy, n (7fc, T^ 0 . Therefore T ( 0 = {'7'j(2^j, 0} is a holomorphic section of F^ over M„ which we denote by ^S{t): ^S{t) = T{t).

(5.42)

^5(0 = 0, namely, ^5,(w„ 0 = 0 means that ^ = Y.x v";{z){d/dw^) e T,{Mt), namely, that ^ is a holomorphic vector field on M^. We denote by H^ the sheaf of germs of holomorphic sections of T{W),: 'E, = €{T{W),). e , = ^(T(M,)) is a subsheaf of S,. If we represent ^ e 0f as ^ = YJI = \ ^fi^/^^f) where Of is a germ of a holomorphic function, the inclusion map t: O , ^ S^ is given by

.:«=i»;^^i(i<.;^R, a=l

OZj

where aw.^/az/ =a(pj^(z,-, t)ldz'f.

A=0 \ « = 1

OZj J dW^-

(5.43.

238

5. Theorem of Existence

If we put ^ e S , in (5.42), we have T{t) = ^S{t)EO{Ft). Consequently ^^r(t) = ^S(t) defines a homomorphism Et-^O(Ft) of sheaves, whose kernel is 0^ as seen above. Thus we have the exact sequence

0->et-^Et^O{Ft)^0.

(5.44)

For a fixed t, choose local coordinates ( z j , . . . , zj) on M, and {wj,..,, wj) on W such that (p](zj, t) = 0, wj = (pj{zj, t) = z/ for a = 1 , . . . , n, and put Sj{Wj, t) = wj. If we write ^G E^ as

then

^=J^t;,^(^)ee,

and

rj(zj, t) = ^Sj(wj, t) = v^

Figure 2

Thus T(r) = ^S{t) is the normal component of ^ to M^. This is the meaning of (5.44). The exact sequence (5.44) gives the exact sequence of cohomology groups • ^ H%M,, E,) -> H'{M,, €{F,)) -^ H\M,,

6,) -> •

(5.45)

§5.2. Number of Moduli

239

Lemma 5.3. The relation between the infinitesimal deformation and the infinitesimal displacement is given by the following. Pt = d'^p,,,

(5.46)

Proof Since U^ = { L^ J is a finite covering of Mt, and each Ujt is a poly disk, d/dteTX^), we see from (5.2) that / / H M , 0,) =//HU,, ®r). For any Ptid/dt) is the cohomology class of the 1-cocycle {Ojk(t)} e Z^iUt, St), where

ot = \

oZj

at

while by Definition 5.5 the infinitesimal displacement is defined as PdA^/^t) = o-(t) = {o-j(Zj, t)}, where o}(z„ t) is given by (5.40). Consequently in order to show that pt(d/dt) = d^^p^jid/dt), by the definition of 5*, it suffices to prove that there is a 0-cochain {^j} with ^j e T{ Up, H J such that

^jS(t) = cTj{zj,t),

^jc-^j = ejM.

Suppose that Wj = gjki^^k) on Wj n W^. Then since ^ : Zj -^ Wj = (Pj{zj, t), we have on Up n U^t
t)),

Zf =ffk(Zk, t).

Differentiating both sides of this equality with respect to t as functions of z j , , . . . , Zfc, t, we have " (9wj^ dffdzk,t) a = idz"

^d(p^(zj,t)_

dt

dt

j^ dw^ ^=odwl

d(pl(Zk,t) dt

Therefore putting

^' xto

dwy

dt

on each Up, we have

A=0 \ a = l

OZj I bWj

Consequently by (5.43) Qik{t) =

^k-^r

On the other hand, differentiating the equality Sj{(pj(zj, t)) = 0 with respect

5. Theorem of Existence

240

to r, we have

11 A=0

dt

dSjjWj, dWj

dt

•Jl= 0,

Wj = (pj(zj,t).

Thus we obtain

mt)- =

o-j{Zj,

t).

We apply the above results to the case in which H^ = P""^^ with n ^ 2 , and Mf is a non-singular hypersurface of order h with h^3. We use the same notation as in (a)(iii) of this section. Let (fo, • • •, ^n+i) be the homogeneous coordinates on W = P""^^ and

a homogeneous polynomial of degree h in fo? • • • ? fn+i- ^t is defined by t{() = 0. Let A be a small polydisk in B = P^ — &. We assume for simplicity that to = 1 on A, and represent a point r G A by its non-homogeneous coordinates ti,..., t^. Thus we have

If we consider t{() as a function of ^ i , . . . , ^^, ^o, • • •, fn+i, the equation t{^) = 0 defines a submanifold of codimension 1 on \y x A, which we denote by M^: M^ = {U,t)eWx^\tU)

= 0}.

The projection W x A -» A induces a surjective holomorphic map tjr: M^ -^ A. Then {M/^, A, tir) is a complex analytic family and for each ^G A, m~\t) = MtXt = Mt is SL non-singular hypersurface on W. The projection Wx^-^ W induces a holomorphic map : M/^^ W. O restricted to Mt = MtXt<::- M^ gives the inclusion M^ ^ W. Thus (J^A, A, TJJ) is a complex analytic family of hypersurfaces on W. Let { W;} be a finite open covering of H^ = P"'^\ and w, = ( w ^ , . . . , wj) the local complex coordinates on W,. We assume that { Wj) and w, satisfy the following: For each j , there is an index ^{j) such that Cpu) ^ 0 on W), and we have w, = (w° . . . , w " ) = ( - ^ , . . . , ^ ^ ^ ^ , ^ ^ ^ ^ , . . . , ^ .

As above we assume that ^A = UJ=i t/jXA, and that
§5.2. Number of Moduli

241

to be given by ^:{zj,t)^wj

=

(pj(zj,t).

Since t{^) is a homogeneous polynomial of degree h in ^o, • • • ? ^n+i, if we put

Sj(Wj, 0 is a polynomial of degree at most h in w],..., gives a minimal equation of M^ in Vl^. Since Sjiwj, t) = Fj,(w)S,{w,, t),

w", and 5,(Wy, 0 = 0

Fj^(w) = ( ^ )

,

(5.47)

\b/3(7)/

the complex line bundle ^t = [^t] depends only on the degree h and is independent of t. Let F be the complex line bundle over W with transition functions Fjj^{w). Then F = [MJ. Further let E be the complex line bundle over \y = P"^^ with transition functions Ej)^(w) = Cpik)/ipu)- Then h times

Let Poo be the hyperplane ofW = P"""^ defined by Co = 0. Then E = [PoJ. Put

Each 5A(^) is a monomial of degree h in ^o,...,

C^+i- Therefore if we put

S;^j(Wj) is a monomial of degree at most h in w ^ , . . . , wj, and on WJ n W^ T^

0, ^A,(W,) = F,-,(>V)5,,(H;,)

(5.48)

holds. Consequently S^ ={Sxj(Wj)} is a holomorphic section of F over W, namely, 5;, G H^(\y, ^ ( F ) ) . We denote by Ft the restriction of F to M^c: W. By (5.39) Ft is defined by the transition functions Ftjk(z) = F}fc(<|)(z)) with z e Up n Ukt- Putting 0-xj(Zj,t)

=

S;,ji(Pj(Zj,t)),

242

5. Theorem of Existence

we have from (5.48) o-xjizj, t) = Ftjk(z)ak(zk, t) on Up n Ukt ^ 0 . Consequently (Jx{t) = {o-;,,(z„ i)} e H\M,, nition 5.5 of the infinitesimal displacement, it is clear that P . . ( ^ ) = ^ ^ = - . W , Lemma 5.4. {cr^{t),...,

€{F,)).

By Defi-

A = l,...,^.

(5.49)

0-^(0} forms a basis of H^(Mt, 0{Ft)).

Proof The sheaf €{Ft) is the restriction of €{F) to M,c: W. Denoting the restriction map by r^, since [ M J = F, we have ker r^ = 0'(F®[MJ"^) = ^. Hence the sequence 0->O^€{F)^OiF,)-^0 is exact. In the corresponding exact sequence of cohomology groups 0 - • H% W, 0) - • H% W, 0{F)) ^ H%M„ €(F,)) -> H\W, H^(W, 0) = C, Sind H\W, 0-^C-^

0)-^---,

€) = 0. Thus we obtain the exact sequence

H%W, 0{F)) ^ H%M„ 0{F,)) -> 0.

For 5x G H% W, 0(F)), we have rtSx = o-M,

A = l,...,/x,

and for S{t) = {Sj{wj, t)}e H\W,€{F)), r,S(t) = 0. Therefore to prove Lemma 5.4, it suffices to show that {S{t), Si,..., S^} forms a basis of H^iW,0(F)). For this it suffices to verify that any s = {sjiwj)}e H^{W,€{F)) is represented by some homogeneous polynomial s{^) of degree /i in ^o, • • •, L+i as 'j(^j) = [ ^ )

-'(O-

(5.50)

For, s{() is a linear combination of t{^), S i ( ^ ) , . . . , 5^(^). Since {s,(w,)} is a holomorphic section of F over W, each Sj{wj) is a holomorphic function on Wj, and on Wj n W^

"'MtJ ,

,

MO)/

s^iw,,)

(5.51)

§5.2. Number of Moduli

243

holds. We haveP"^^-Poo = C"'^\ and putting z« = f,/^o for a = 1 , . . . , n + 1 , we have complex coordinates ( z j , . . . , z„+i) on C"'^^ Since by (5.51) on W;-n ^fc n C"^ V 0 , we have

there is a holomorphic function giz^,...,

z„+i) on C""^' such that on each

g(z„...,z„,,) = ( ^ )

-sjiwj).

(5.52)

Let oo

g(zi,...,z„+i)= X

gk(zu...,z^+i),

fc = 0

where gfc(zi,..., z„+i) is the homogeneous part of degree k of the power series expansion of g ( z i , . . . , z„+i). Take an arbitrary point p = (0, ^ 1 , . . . , L+i) on Poo. Then ;? belongs to one of W/s. Fix f i , . . . , f„+i for a while. If we take E > 0 sufficiently small, then for 0 < |^o| < ^, we have

\

ho

bo /

Therefore by (5.52) we have ^0 I fc-0

=^PU)'SjW-

Jk ^0

As a function of ^o, the right-hand side is holomorphic on |fo|<^- Consequently we must have gkUu • • •, ^n+i) = 0 for k^h-^l. Since ^ i , . . . , ^„+i are arbitrary, g ( z i , . . . , z„+i) is a polynomial of degree at most h in Z i , . . . , z„+i. Therefore if we put

\bO

bO I

5(f) is a homogeneous polynomial of degree /i in fo? • • •, f«+i, and (5.50) follows from (5.52). I We consider the exact sequence (5.44): 0^e,^H,->(^(F,)^0

244

5. Theorem of Existence

for M, c: W = P " ^ \ Let S = 0{ T( W)) be the sheaf of germs of holomorphic vector fields over W. Then H^ is the restriction of H to M^. Let r^ be the restriction map. Then since [M,] = £ ^ ker rt = E®E~^ = OiT{W)®E~^). Thus 0 - • H® E~^ -> S -X H, -> 0

(5.53)

is exact. Lemma 5.5. Except in the case n = 2, h = 4, we have H^(P"''^ H® £-^) = 0,

^ = 0, 1, 2.

Proof. Put T= TCP"""'). Then S x £ - ' = 0{T®E-''), of T®E-'' is T*® JB^ Hence by (3.155),

and the dual bundle

dim H^(P"^^ H ® £ - ' ) = dim //"^^-^(P"^\ a " ^ H ^ * ® ^ ' ) ) . Putting K = A"""' T*, we have n''^\T''®E'')

= €(T''®K®E'')

=

n\K®E''),

while the canonical bundle K of P"^^ is iB"""l Therefore dim

JF/^(P"^\ S ®

£ - ' ) = dim H"^^-^(P""'\ OH^'"""^)).

(5.54)

Since we assume that n ^ 2 and /i ^ 3, Lemma 5.5 follows immediately from following Bott's theorem. I Theorem 5.2 (Bott [1]). / / ^ ( P " ^ \ a ^ ( £ ' ' ) ) vanishes except for the following cases: (i) p + q and /c = 0, (ii) q = 0 and /c>/?, (iii) g = n 4-1 and A: < 0 - n - 1. Thus except for the case n = 2, /i = 4, in the exact sequence of cohomology groups induced from (5.53): 0-^ H\W,E®E-'')-^

H%W,E)^

H\M,,E,)

->H\W,E®E'-^)-^"' we have H^W,

S ® £"^) = 0 for ^ = 0,1, 2. Hence H%W,E)^H%M,E,),

and

(5.55) H\W,E)

=

H\Mt,E,).

§5.2. Number of Moduli

245

Also from (5.54) and Theorem 5.2, we have H\P"^\

S) = 0. Hence

An automorphism of \ ^ = P"'^^ is a projective transformation, and the set ® of all of them forms a Lie transformation group of W with dim @ = (n + 2 ) ^ - 1. Since H^{ W; S) is the Lie algebra consisting of the infinitesimal transformations of @, we have dim H\W,E)

= {n-\-2f - I.

Hence from (5.55) we have dim H\M,,

S,) = (n + 2 ) ^ - 1.

(5.56)

In the exact sequence of cohomology groups induced from the exact sequence 0 ^ 8^ ^ H^ ^ ^(Ft) -^ 0, we have H\Mt, S^) = 0. Also we can prove that H^(Mt, St) = 0 (see [21], p. 406, Lemma 14,2). Therefore the sequence 0-> H%M,, S,) ^ H^(M, 0{F,)) -> H\M,,

6,) -> 0

(5.57)

is exact. By Lemma 5.4, dim H^{M,, ^(F,)) = /JL. Therefore by (5.56) d i m / f H M „ 0 , ) = m = / x - ( n + 2 ) ' + l = ( " ' ^ ^ ^ J - - ( n + 2)^ Thus we obtain (5.21). For a projective transformation ye®,

define

Then y: t^yt is a projective transformation of the parameter space P^. Since S,( w,, t) = (1/f/30))^^(f), if 7 belongs to a sufficiently small neighbourhood of the identity of @, we have Sj{wj,yt) = Sj(y-'wj,t)

(5.58)

for WJEWJ and t = {\, ti,..., t^)e^. We denote by X L i ^A(^) ^/^^A the infinitesimal transformation of A determined by the infinitesimal transformation ^e H^iW, H) of ®. Since f is a holomorphic vector field on W, ^ is written on each WJ as ^ = ZA=O ^J(^j) d/dwf. From (5.58) clearly we have

A=i

dh

246

5. Theorem of Existence

Substituting Wj = (pj(Zj, t) and denoting the restriction of ^ to M^ by ^^G H^(M„ H,), we obtain from (5.41) and (5.42) the equality — ]=^t^i^)' 'A'tj

Z Ut)PdA A= l

(5.59)

This equality implies that if Zr=i ^ A ( 0 ^/^h = 0 for a point r G A, we have ^tS{t) = 0. On the other hand, f, ^ ^,(5(0) gives the injection H^(M„ H,) -^ H'^iM,, €{F,)) in (5.57). Therefore ^,S{t) = 0 implies ^, = 0, hence by (5.55) we have f = 0. Thus if ^ 9^ 0, I^^^ ^^(0 ^l^h ^ 0. This proves that for any te A, the map y-^ yt is a biholomorphic map on a sufficiently small neighbourhood of the unity of ®. Fix an arbitrary point t^ e A, and take a sufficiently small neighbourhood U of the unity of ®. Then y^ yt^ maps U biholomorphically onto U/^c A. U f ^ is a domain of a submanifold of dimension (n + 2)^ — 1 of A containing t^. Since /jL = m + (n+2)^—l, rearranging ti,..., t^ if necessary, we may assume that the coorcjinate plane tm+i = ^m+i, • • •, ^^ = ^^ through t^ intersects with Ut^ transversally at t^. If we take a polydisk with centre t^: t / = { ( ^ 1 , . . . , tm. tl^u

• > • , tl)\\t,-

t'llK

e, , . , ,\tm-

tl\<

S},

the map (% 0 ^ yt maps UxU biholomorphically onto the neighbourhood U t/ cz A of f^ provided that e > 0 is sufficiently small. Consider the restriction (Mu, U.vj) of {M^, A, XET) to t/ c A, where Jtu = tu~\U). Since 111/ is biholomorphically equivalent to UxU, we have TXA)=T,{nU)=T,(nt)^T,{U) at each point te U. Let e be the unity of ®. Then the tangent space Te(U) of U at eGU is H\W,E):

TM)

= H%W,E).

Therefore T,(nt) is isomor-

phic to H%W, S). The isomorphism H%W, E)-^T,{Ut)

is given by

Ot;^

A= l

By (5.59) we have PciAv^) = ^tSit). From this and (5.46), (5.57), we see that the diagram 0

> H%M,,E,)

> H\M,,€{F,))

-^H\M,,%,)

•O (5.60)

^^"""^^

0—*

//"(W.H)

— U r,(A)

pf

§5.2. Number of Moduli

247

is an exact commutative diagram. In this diagram, p^ ^ is an isomorphism by (5.49) and Lemma 5.4, r^ is an isomorphism by (5.55) and r]H^{ W, S) = TtiUt). Consequently since Tt{^)= T,(nt)-hTt(U), p, maps T,(l/) isomorphically onto H\Mt,St). Thus the complex analytic family {Mu, U, m) is effectively parametrized and dim H\Mt, St) = ^- Therefore the number of moduli m(Mt) = m of each M^ = V7~\t) for te U is equal

to dim H\Mt, St). Since t^ is an arbitrary point of A, and A <= B is an arbitrary sufficiently small polydisk in B, we have proved that except for the case n = 2, /i = 4, the number of moduli m(Mt) is defined for any non-singular hypersurface M, of order h in P"^^ and that m{Mt) = dim H^M^, ©,). In case n = 2, /z = 4, as stated before, we have m = 19 and dim H\Mt, @ J = 20. Also in this case we have the following exact commutative diagram similar to (5.60) 0 - ^ H^(M„ S,) —> H\Mt,

0—>

OiFt)) -^H\Mt,

6,) - > H^M^, H,) —> 0

H\W,E)

But H^{Mt, S^) 7^ 0 in this case. Since in the exact sequence of cohomology groups induced from (5.53) we have H\P^, H) = H^{P^, S) = 0, we obtain H\Mt,Et)^H\p\E®E-''), Consequently by (5.54) dimffHM„H,) = d\mH\p\[l^), while H\P\n^) belongs to the exceptional cases (i) of Bott's theorem, and actually we have dim H^{P^,Ct^) = l (see p. 175). Therefore dim /f'(M„ e,) = m + 1 = 20. At first we could not find the reason why the case n = 2, h = 4 is exceptional. Then S. Nakano, who stayed at the Institute for Advanced Study at Princeton at that time, pointed out that in this case the complex analytic family (M, B^m) is not complete. Actually taking the hypersurface ^o + ^?+ ^2+^3 = 0 of Fermat type as M^, he showed that M^ is an elliptic surface and, using the theory of elliptic surfaces ([14]), proved that there is a complex analytic family {7V„|MGC} with No= Mf of deformation of M, consisting of elliptic surfaces such that for any given e > 0, we can find a non-algebraic surface N^ with |M| < e in this family. From this result, we considered that the reason why the number of effective parameters m = 19 of the complex analytic family {M, B, m) for a quartic surface in P^ is less than dim H^(Mt, B^) = 20 would be that M does not contain all possible deformations of M^, and so we expected that the number of moduli m(Mf) of M^ could still be defined in this case and equal t o d i m / / ' ( M „ @ , ) = 20. Next we studied hypersurfaces on Abelian varieties ([21], §14(6)). We explain them briefly below. Let A = A""^^ be an (n +1)-dimensional Abelian variety with n ^ 2 . A is a submanifold of some projective space P^. For a

248

5. Theorem of Existence

general hyperplane P of P^, M = A n P is a non-singular hypersurface on A Given a complex analytic family {Mt\te B} of this hypersurface MQ = M with 0 G J 5 C : C ' " , there exists a complex analytic family of deformations {At I ^ G A} of Ao = A with M^ c: A^ provided that A is a sufficiently small polydisk with 0 G A c: R A^ is a complex torus, and M^ ci A, implies that At is an Abelian variety. The number of moduli of Ao = A"^^ as a complex torus is equal to (n +1)^, but if we only consider deformations of AQ which are Abelian varieties, its number of moduli reduces to (n + l)(n + 2)/2 since their period matrices must be Riemann matrices. In view of this fact we suspected that the number of moduli m{M) of M would be smaller than dim//^(M, 0 ) , but the computation showed that m(M) = dim//^(M, 0 ) also in this case. Of course the decrease of the number of moduli of At does not necessarily imply m(M) < dim//^(M, B), but at that time this result seemed surprising to us.

§5.3. Theorem of Existence In the preceding section we have seen that for several examples of compact complex manifolds M, the number m{M) of moduli of M is equal to dim//^(M, @). As stated in (a) of the preceding section, if m{M) = dim f/^(M, 0 ) , for an arbitrary ^ G //^(M, 0 ) , there exists a complex analytic family {M, B, W) with OG 5 c C satisfying the following conditions: m-\0)

= M,

i—rA \ dt

=0,

where

Mt = v7-\t).

(5.61)

/t^o

Since at that time we could not find no example M with m{M)?^ dim //^(M, 0 ) , we expected that for any given compact complex manifold M and any de H^{M,S), there would exist a complex analytic family (M, B, m) satisfying (5.61) but for some exceptional cases. (a) An Elementary Method We first tried to prove the theorem of existence by an elementary method, using power series expansion. We state it in this section, for, although we did not succeed in proving the theorem by this method, this was an important step in the development of the theory of deformation. In order to prove the theorem of existence, it suffices to take as J? a disk A = {r G CI U| < r} of sufficiently small radius r > 0, and construct a complex analytic family of the form (4.27) /

§5.3. Theorem of Existence

249

satisfying (5.61). Here each Uj is a polydisk in C" and Ji is the complex manifold obtained by glueing the domains U, x A, 7 = 1 , . . . , / , by identifying {zj, t) eUjXA and (z^, t)eUj,xA if zf =fjUzk, t). fjUzk, 0 = ffk(z\,..., Zfc, 0, a: = 1 , . . . , n, are holomorphic functions defined on the open subset U^ x A n L^ x A of the domain 14 x A of M. We call these holomorphic functions/^^(z^, t) the defining functions of M. These functions must satisfy the following compatibility conditions: On (7^ x A n L^ x A 7^ 0 , frjc(z,,t)=f^(fj^(z,,t),t).

(5.62)

Since by (5.61) M = m~'\0) = U]=i Uj xO, identifying Uj with Uj xO, Uj is considered to be a coordinate polydisk on M and Zj = {z],..., z") are considered to be local coordinates of M defined on Uj. If we expand/,^(zfc, t) into the power series in t as 00

mzu,t)=lfrk\A^^K,

(5.63)

t/ze coefficients f^\j,(zk) are holomorphic functions defined on the open subset UknUj9^0 of the coordinate polydisk ^4 on M. On L^ n Ui^= UjXOn t/fcXO, we have z" =/^^|o(zfc). Thus z" =/,^|o(zfc) gives the transformation of local coordinates of the complex manifold M Therefore we may consider that the coefficient f^k\o(^k) of t^ in (5.63) is given with M. In order to construct a complex analytic family Ji = Uj=i ^^^ with tiT~^(0) = U^._i U, xO = M, it suffices to give the defining functions/j^(zfc, t) of M. For this, since/j^|o(zfc) =f^ki^k, 0) are already given, we only have to determine the coefficients ffk\v{zk), ^ = 1 , 2 , . . . , of (5.63). The defining functions of M satisfy the equalities (5.62): fUzk. t) =frj(fjk(Zk, 0, 0 ,

a - 1 , . . . , n.

Hence the coefficients f]l\^,{zi^), ^ = 1, 2 , . . . , must be determined such as 00

fMzk,t)=ffk\oiZk)^ I frkUzk)t^ satisfy these equalities. For a given cohomology class 6 e H^(M, 0 ) , take an arbitrary 1-cocycle {Oj}^}eZ^{{Uj}, ©) belonging to 0, and write Oj^ in the form

250

5. Theorem of Existence

Since Z I ^

'

I

7~^= I /jfcliC^k)!-^,

if we put /jk|l(^k) = ^jk(Zk),

Zj

the infinitesimal deformation of M^ = U^.^j ^^t

=fjk\o(Zk),

is by the definition (4.15),

/dM\

\ dt A=o Therefore in order to construct a complex analytic family ^ = U^^i ^^^ satisfying (5.61), it suffices to put fjk\i{zk) = ^fk(Zj) first and to determine fjicU^k), ^ = 2, 3 , . . . , such that they satisfy (5.62). For simplicity, putting fjkiZk, t) = (fjkiZk, 0 , . • • ,fJk(Zk, 0 ) , and fjk\v{Zk) = {fjk\viZk), . . . ,fJk\AZk)), and using vector notation, we write (5.63) in the form oo

fjkiZk,t)=lfjkUzk)t\

(5.64)

In general for a power series in t oo

p ( 0 = Z p . r = Po+p,f+---, we put p-(t) = Po-^Pit+- • • + P , r . Further for two power series P{t) and Q(t), we write P{t) =^Q{t) if P ( r ) ^ Q ( 0 ( m o d r ' " 0 . Thus P(r) = , Q ( 0 just means that P " ( 0 =
^ = 1,2,....

(5.65).

§5.3. Theorem of Existence

251

As Stated above, yjfc|o(^k) are given with M. Also we have putyjfc|i(^fc) = 6jk{zj) with Zj=fjk\o{zk)' Then we can verify (5.65)i easily as follows. f]jifjk(Zk, t), t)=f]j{fjj,lo(Zj,)+fjj,li{Zk)t, =

t)

fy\o(Zj+fjk\liZk)t)-^fij\l(Zj-\-fjk\i(Z},)t)t n

= fij\o(Zj)-^ 1

Q

Z 13 = 1 dZj

—pfij\o(Zj)'fjk\l(Zk)-^fij\liZk)t,

^NhilQ fijio(Zj) =fik\o{zk)' Hence if we put zf =/^|o(z,), (5.65)i is equivalent to the equality

frk\i(zk)=fh{zj)+ i

^^ffk\i{zk\

namely, n

^

n

Z fnc\i{Zk)—^= a=l

OZi

d

^

d

Z f?j\iiZj)—^-^ Z ffk\iizk)—-p' a=l

OZi

13 = 1

dZj

If we put^fcii(zfc) = djk(Zj), the last equality follows from the fact that {^^^(^j)} is a 1-cocycle. We consider the vector y;fc|^(zfc) = (/]fc|^(zfc),... ,/Jfc|^(zfc)) as the holomorphic vector field on Uj n Uk'.

fjk\i^ =

Z fjk\i^(Zk) a=l

T~^ OZj

if necessary. On each UknUjT^ 0 , we determine fjk\Azk), ^ = 2, 3 , . . . , by induction on p. For this assume that on each Ukn Uj9^0, fJk\Zk.

t)=fjklo{Zk)+'

'

•+fjk\.-l{Zk)t''~'

is already determined in such a way as the congruence (5.65) ^_i

/r'(z., t) - fr\fj^\z,„ t), t) v—l

holds and consider (5.65)^:

fik(Zk,t)^f-jifJ,iz„t),t). For simplicity, we write fjkiz^, t), fjk\Azk), etc., as /JfcCt), ^ | ^ etc., respec-

252

5. Theorem of Existence

lively. Then the right-hand side of (5.65)^ is written as

=fv\rjk\t)+fMX,t)+MAfww. Putting Zj + {, = {z] + f ] , . . . , z" + ^j') and expanding each component of the vector/|J~^(z; + f^, t) into the power series in ^ ] , . . . , fj', we obtain

Putting zj=fjk\t) Tlnfij dZj

and Cj=fjk\J\ since ifjk

( 0 ? ^jfj^v^""

= T~Bfij\0\fjk\o)fjk\v^ ^ dZj

•>

we have

/r'(/jrHo+y;.i.f'',o=/r'(/;r'(o,o+ i A/>io(y;
On the other hand, clearly we have

filUfJk{t))t''^f,lAfMo)t''=fyUzj)t''. V

Therefore (5.65)„ is reduced to

^^1dZJ

Since by hypothesis (5.65)j,_i holds, the power series expansion of the left-hand side of this congruence begins in the term of degree v in t. If we put the coefficient of t"" as ryfc|^(zfc), we have f\k\zk. t)-nr\f-\z^,

t), t) - r^.,|.(z,)r,

(5.66).

§5.3. Theorem of Existence

253

hence, (5.65)^ is reduced to the equality " dz^ijk\Azk)= Z T-^ffkw(^k)-fik\A^k)-^fij\AZj)'

(5.67)^

j8 = l OZJ

Here Zj,, Zj = fj^ioi^k) and z, =yy|o(z,) =/fc|o(zfc) are local coordinates of one and the same point of M contained in U^n Ujn Uf. Each component r^fc|^(zfc) of the vector Tyki^iz^) is a holomorphic function defined on the open set t/jO UjnUk7^0 of M. The equaUty (5.67)^ is written in terms of the components as Ttjk\Azk)= 1 —^ffk\.(Zk)-ftk\.{zk)+fy\Azj),

a=

l,...,n.

Using vector field notation, we can write it in the form

Zr^fci^(zfc)—^=X/ffc|v(zfc)—^-I/r/c|v(^fc)—^+Z/?!^(^/c)7Ta

dZi

p

dZj

OZi

ct

a

OZi

Therefore putting Ty-fcli. = Z r^fc|l/(^k) T~^5 a

OZi

fjk\0 = llfjk\p\^k) /3

7~flJ • • • ? OZj

we can write (5.67)^ as ^ijk\v=fjk\v-fik\v-^fij\v' (5.68)^ Thus if (5.65)^_i holds, (5.65)^ is equivalent to (5.68)^. If we consider {fjk\v} as a 1-cochain on the finite open covering U = { Uj] of M, the right-hand side of (5.68)^ is equal to its coboundary 8{fjk\^}. In our case, however, although^;y|^ = 0, ^ ^ 1, sinceyj^(zy, t) = Zj are independent of ^,fkj\p =fjk\u ^o not necessarily hold. In view of this we extend the definitions of l-cochains, 2-cocycles, etc., as follows. Let 5^ be an arbitrary sheaf over M. Suppose that for each pair j , k of indices with L^ n [4 T^ 0 , a section a}^ e r ( U, n U^, 5^) is assigned. If CTJJ = 0 for every j , the set c^ = {o}^} of these o-j^ is called a 1-cochain on U = { Uj}. Similarly suppose that each triple i,f k with Utn Ujn 17^9^0, a section o-yfc G r ( L/j n Ujn L4, S) is assigned. If au^ = cTij^k = 0, and on each Uhn UiO

UjnUk9^0 <^ijk ~ <^hjk + ^hik ~ ^hij — 0

holds, the set c^ = {c7yfc} is called a 2-cocycle on U. We define the coboundary of a 1-cochain c^ = {0}^} as ^c^ = {'^i/fc},

Tijk = o-jk = o-jk - o-ik + o-ij.

254

5. Theorem of Existence

It is easy to verify that the cohoundary 8c^ is a 2-cocycle. Let C^(U, Sf) be the Abelian group of all 1-cochains c^ and Z^(Vi, 5^) that of all 2-cocycles c^. Then 6C^(U, 5^) is a subgroup of Z^(U, 5^). The quotient group

H\n, ^) = z\n, s/')/8c\n, ^)

(5.69)

is called the 2-dimensional cohomology group of U with coefficients in 5^. With this notation, (5.68)^ is written as {Tijjc\.} = HfM.h

(5.70).

Lemma 5.6. {Tyki^} is a 2-cocycle on 11: {r,-,|jGZ^(U,e). Proof. Using vector notation, we must prove that ll~~^^fjk\j^{zk) -^ hjk\v(Zk) -^T hik\A^k) -^ hij\A^j) = 0.

(5.71)

Here we put z^ = ( z j , , . . . , z^), and Zh = fhk\o{Zk)- For simplicity we write ^hijk\., rhk\t), etc., instead of Thjk\.izk), fhk\zj,, t), etc., respectively. By (5.66). we have ^hik\vt'' =fhk {t)-fhi

(/iTc" ( 0 , 0 ,

T,MX=rkk\t)-rki\fjk-\t\

t),

hence, r,.fc|.r-r,,,|.r'' =/;;r'(/-'(o, o-/hr'(/,T'(0,0Substituting Zj =fjj^^(t) into the congruence

we obtain

Further since

§5.3. Theorem of Existence

255

we have

where Zh = fhi\oifv\oifjk\o(zk)))

=fhi\o{Zi)- Hence

a^lOZi

Therefore we obtain

a = l oZj

Thus (5.71) holds.

I

We denote the cohomology class of the 2-cocycle {Tyki^jeZ (U, 6 ) by r^ G H^(Vi, @). As stated above, when/Jfc"^(zfc, 0 satisfy (5.65)^_i, the system of congruences (5.65)^ is equivalent to the equality (5.70)^: {Ty^i^} = 8{fjk\r,}. Therefore in order to determine/Jfc(^fc, 0 such that (5.65)^ holds, it suffices to determine the 1-cochain {fjk\v} e C^(U, O) such that S{fjk\^} = {Tijk\^}. First suppose r^ = 0. Then if we take an arbitrary 1-cochain {CTJ^} e C\U, B) with {ryfc|^} = 6{o)fc}, and put ^fc|i/= ^jfc, (5.65)^ holds. If F^^O, there exists no 1-cochain {o}^} with {Tijk\^} = 8{ajk}, hence we cannot determine {fjk\u} such that (5.65)^ holds. In this sense we call F^ the obstruction to deformation of M We always have Ty-fcii =fik\oiZk) -fij\oifjk\o{^k))

= 0,

hence Fi = 0. Therefore we call F2 the first obstruction, and F^+i the ^th obstruction. Here note that F^ is not defined unless (5.65)^_i holds, since {^ijk\p} is defined by the congruences (5.66)^

In other words, unless F2 = 0 , . . . ,F^ = 0, F^+j is not defined. Moreover even if F2 = 0 , . . . , F^, = 0, Fj,+i depends in general on the choice of {fjk\v} with 8\fjk\r.} = {Tijk\A, hence, it maybe that F^+j 9^ 0 for one choice of {fjk\j.}, and F^+i = 0 for another. Thus it is very difficult in general to see whether we can determine {fjk\v} such that the congruences (5.65)^ hold for all ^ = 2,3-.--

256

5. Theorem of Existence

In case H^{U, 0 ) = 0, however, since Z\U, 0 ) = 8C^{Vi, S), we can determine 1-cochains {fjk\i.} successively for p = 2,3,... in such a way as 5{yjfc|^} = {^ijk\i^}' Thus we can determine power series oo

/jL(zfc,0= I/j)c|.(zfc)r v= 0

satisfying the equahties (5.62): fUzj., t) ^fUfjkizk,

0, 0,

a = 1 , . . . , n.

If ail these ffki^k^t) converge, then on a sufficiently small disk A = {r G C| Ir| < r}, ffk(zk, t) are holomorphic functions on (L^ n L4) x A, hence, the existence of a complex analytic family (Ji,B,vT) satisfying (5.61) is proved. But there are infinitely many choices of 1-cochains {fjk\u} with ^{fjk\j.} = {Tijk\v} for each z/ = 2, 3 , . . . , and ffki^k^t) do not converge in general for arbitrary choice of {fjk\v}. Thus we have to show that the power series fjk(^k,t)=

i

fJk\AzkW

converges if {fjk\v} are suitably chosen. But for all our efforts, we could not prove the convergence. Thus we did not succeed in proving the theorem of existence by an elementary method. Although we failed to prove the theorem, we proved that, in case H^(U, 0 ) = 0, there exist formal power series fjk(Zk,t)=

I

fjk\v{Zk)t"

in t satisfying the fundamental equations/^(^/c, t) — fijifjk^^k, 0? 0- As will be shown below, H\M, 0 ) = 0 implies H^(ll, 6 ) = 0, hence, ifH^iM, 6 ) = 0, the fundamental equations fikizk, t) = fj{fjk{zk, t), t) have formal power series solutions. This gave some hope that, if /f^(M, 6 ) = 0, there might exist a complex analytic family (Jt, A, m) satisfying the conditions (5.61): m-\0)

= M,

(-jT-^)

=0,

(^A=o~''

M, =

m-\t)

foiany eeH\M,e). We now give proof of the fact that H\M, e) = 0 impHes H^{U, @) = 0. Proof By a local C°° vector (0, q)-form we mean a local C°° section of the vector bundle T{M)®/\^ f'^(M) over M where T{M) is the tangent

§5.3. Theorem of Existence

257

bundle of M A C°° vector (0, ^)-form ip is written as ^ = l L i ^Ji^/dzJ) where ifjj are C^ (0, ^)-forms. We denote by ^ ^ ' ^ ( r ( M ) ) the sheaf of germs of C°° vector (0, ^)-forms over M. Since (d = €{T{M)), by (3.106) we have

H\M, @) ^ r(M, 5^^''( r(M)))/ar(M, j?/^'^ T(M))).

(5.72)

Let {pj} be a partition of unity subordinate to U = { Uj], Take an arbitrary 2-cocycle {^yfe}GZ^(U, @). By the definition of a 2-cocycle, we have ^ijk — ^hjk ~ ^hik + ^hij-

Multiplying by ph, and taking the summation with respect to h, we obtain ^ijk — ^jk ~ ^ik + ^ip

^jk — Z Ph^hjkh

Each ^jk is a C°° vector field on l/jnUk?^ 0 . Since a^y^ = 0, we have d^jk =

d^ik-d^ij,

hence d^jk = ^ k -

^p

^j = Z

Pid^ir

Here each i/r^ is a C°° vector (0, l)-form on Uj. Since 5i//[/ = dil/k on L^ n L4J there is a_5-closed C°° vector (0, 2)-form cp e T{M, dsd^'\T(M))) on M such that (p = dif/j on each U,. Since H^{M, ©) = 0 by assumption, by (5.72) there is a C°° vector (0, l)-form if/ e r ( M , J2/°''(T(M))) such that ^ = dil/. We have d{il/j - ij/) = 0 on each L^, while ^ is a polydisk. Therefore by Dolbeault's lemma (Theorem 3.3), there is a C°° vector field rjj on U; with ^J-\IJ = drjj. Then since 5^jfc = ^k-^j

= dVk-

drjj,

putting ^jfc = ^7k -Vk-^

Vp

we have 56>yfc = 0, hence Oj]^ is a holomorphic vector field on Uj n L4. Moreover ^^y;, = 0 implies |y = 0, hence Ojj = 0. Clearly we have ^ijk — ^jk ~ iik + ^ij == ^jk ~ ^ik + ^ifo

which implies that {^y7c} = 5{6>fc}, with 0. I

{(9,JG

C'(U, ©)• Thus H^(U, 0 ) =

In §5.1, we define the primary obstruction [6, 6], We have [0, ^] = 2r2.

258

5. Theorem of Existence

Proof. Since Z^(U, e ) c : Z^(U, ®), C^U, 0)c= C^U, 0 ) , and, as is easily seen, Z'(U, 0 ) n 5C'(U, 0 ) = 8C\Vi, 0 ) , we have //'(U,0)^H'(U,0). Consequently in order to prove that {6, 61 = 2^2, it suffices to show the existence of a 1-cochain {TJ^}^ C ^ ( U , 0 ) such that

By (5.66)2, we have

hence ryfc|2(^k) is the coefficient of t^ in -f]jif]k(^k, t), t). Since fljifjkit),

t)

=fij\o{fjk\Q-^fjk\\t)+fij\l{fjk\0^fjk\lt)U

putting Zi =yy|o(z^), Zj=fjk\o{zk), we have ^ ijk\2-

^ L z p^y^i

^B^^yJjk\\Jjk\\ dZj oZj

L ^BJjk\\' p = i dZj

Therefore, since fjk\i ~ ^jk-> we have 2ryfc|2 = - Z ^JkOfk _ y ' fi ~2^jfc • ^y

\ p dZJ

I

13 dZj

while Xp (5z,./azf )0ffe = e,fc - 0,y. Hence 5z2ryfc|2 = Z T~;8 ^Jfc ' ^jk ~ ^jk ' ^ik - ^jk ' ^ij' p dZj

Since [(9y, Oj^] = [% 60,] = Oy • Oik - Oij, • % we have dz^^ijk\2

~ [ ^ip ^jk] = Z r~J8 ^7fc * ^jfc ~ ^ik ' ^ik + ^i/- • ^ijP dZj

Thus putting TJJ, = Z^ ^jk' Ofk{d/dzf) on each L/j n 11^9^0, we obtain

§5.3. Theorem of Existence

259

Let N be a 2-dimensional complex torus. Then the product M = P^ xN gives an example for which there exists a 6e H^{M, S) with [6, 0]7^0. If the number of moduli in the sense of Definition 5.4 is defined for this M = P^ xN, we would have m ( M ) < d i m H^(M, 6 ) since there is a ^ G H\M, 6) with [^, ^] 7^ 0. But for the number of moduli m{M) to be defined for M, there must exist an effectively parametrized and complete analytic family {M, B, m) with 0 e B c: C'" such that m~\^) = M. For M = P'xN, such a family does not exist. This is verified as follows. Suppose that such a family (M, B, m) exists. The set of the infinitesimal deformations po{d/dt) = {dMt/dt)t=.o of Mt = v7~^(t) at t = 0 forms a vector space po(To{B))c: H\M, e ) . As stated in §5.2(a), if 0, rj e po( To(B)), [0,rj] = 0. Consider two complex analytic families (Jix^^^'^x) with tzr^^(0) = M and A^ = {5 G C11^1 < r} for A = 1, 2, and let 9^ = {dm^^{s)/ds)s=o be the corresponding infinitesimal deformations at s = 0. Since {M,B,m) is assumed to be complete, taking a sufficiently small disk A^ with r > e > 0, we may assume that both (tn-A^(Ae), A^, mx),\ = 1,2, are complex analytic families induced from {M,B,m). Therefore by (4.30), e^epo(To(B)) for A = 1,2, hence [^i?^2] = 0- But for M = P^xN, there exist complex analytic families iJt;,,^^m^), A = 1,2, with m^\0) = M such that [^1, ^sJ^O ([21], §16). Thus the number of moduli m(M) is not defined. At that time we had no example M of compact complex manifolds whose number of moduli m(M) is defined and yet m{M) 9^ dim H^{M,S). In view of this we made the following conjecture. Conjecture. Let M be a compact complex manifold. If the number of moduli m{M) of M is defined, m(M) is equal to dim H^{M, B). Our paper "On deformations of complex analytic structures, I-IF', often quoted already in this book, treats various results obtained up to that time centring on this conjecture. Given a compact complex manifold M, it is in general very difficult to compute the number of moduli m(M) or dimff^(M, 6 ) . Consequently from the mere fact that m{M) = dim JF/^(M, O) always holds for several simple types of compact complex manifolds M for which we can compute m{M) and dim //^(M, O), we cannot conclude the universal validity of this equality m{M) = dim H^(M, 0 ) . But the above conjecture has actually turned out to be very useful as a "working hypothesis" in our study of deformations. (b) The Theorem of Existence Since we did not succeed in proving the theorem of existence by an elementary method, we—Spencer and the author—tried to prove it by another method. Let (M, B, m) be a complex analytic family of compact complex manifolds, and put Mt = m~^{t) where B is a domain of C^ containing the origin 0. Define \t\ = maXxl^Al for r = ( ^ j , . . . , ^^)G C", and

260

5. Theorem of Existence

let ^ = ^r = {teC"'\\t\ 0 . If we take a sufficiently small Ac: 5, Jt^=wr~^{A) is represented in the form (4.27): ^A = U UjXA.

(5.73)

j

We denote a point of Uj by {/ = ( ^ ] , . . . , ^J). For simplicity we assume that Uj = UjeC'^\\^j\
te^.

Moreover, as is clearly seen from the proof of Theorem 2.5, ^ is the identify of M = M xO onto M = MQ^M^, namely ^ ( z , 0) = z. Put ^ ( z , 0 = {L 0 = ((,, t) for ^ ( z , t) e Uj X A. Then each component CJ = ^/(^, t), a = l,,.. ,n, of ^, = ( ^ ] , . . . , ^;) is a C°" function:

nz,t)=u]iz,t),...,cj(z,t),t,,..,,tj. If we identify Jt^ = '^{M x^) with M x A via "^.Jt^ is considered as a complex manifold with the complex structure defined on the C°^ manifold M x^ by the system of local complex coordinates { ( 4 t)\j=1,2,3,...},

UJ, t)=(^](z,

0 , . . . , ^;(^, 0 , ^ 1 , . . . ,

tj.

Consequently for each t, M^ is the complex structure of the differentiable manifold M defined by the system of local complex coordinates: {z^ijiz,

0|7 = 1,2,...},

{,(z, t) = Uj{z, 0 , . . . , m^.

t)).

Here by "the differentiable manifold M " we mean the underlying differentiable manifold of the complex manifold M.

§5.3. Theorem of Existence

261

If we describe {M^, A, m) as above, each component of local complel coordinates ^^(z, t) is represented as a local C°° function ^/(z, t) on M depending on t, and the underlying C°° manifold M of M^ appears explicitly as the place on which ^/(z, t) are defined. M is itself a complex manifold. Let ( z \ . . . , z") be arbitrary local complex coordinates of a point z of M. f;(z, t) = Cf{z\ . . . , z", ^ 1 , . . . . , t^),

a = 1 , . . . , n,

are C°° functions of the complex variables z \ . . . , z", ti,...,t^. Since for r = 0, both (^](z, 0 ) , . . . , ^"(z, 0)) and ( z \ . . . , z") are local complex coordinates on the complex manifold MQ = M, ^/(z, 0) are holomorphic functions ofz^,..., ^^ and

de.(5^) holds. On the open set UjXtn

.0

(5.74)

U^xt of M^, we have

and det(a^//a^^) 9^ 0. From this and (5.74), if we take A sufficiently small, it follows that det(<%^) \

dZ

^0

(5.75)

I a,\ = \,...,n

for any ^ G A. We denote the tangent bundle of the complex manifold M by T: T= T{M). Also we denote by if^'^(T) the linear space of all C°° vector (0, g)-forms, namely, C°° sections of T®/\^ f* over M : if^'^(r) = r ( M , ^^'^(T)). cpe cS^^'^(r) is represented in the form « (P=l

^

^P^TT. A=i 5z

1 (p''= — g!

y(p^,...f>^(z)dz''^A"'Adz\

where (p^j...j5^(z) are local C°° functions on M which are skew-symmetric in the indices Pi,..., Vq, For simplicity we denote d/dz^ by d^, and d/dz^ by d^. Consider 5^7 (z, t) = ILjvCJiz. t) dz\ The domain % = ^-\Uj xA) of ^/(z, t) is a domain of M X A. Since by (5.75), det((9A^/(z, t)) ¥^ 0, there is a unique (0, l)-form

1^=1

262

5. Theorem of Existence

for each A = 1 , . . . , n, such that dCj(^^ 0 = 1

cpfiz, t)d,CJ{z. 0 ,

a = 1 , . . . , n.

(5.76)

The coefficients (pp{z, t) are C^ functions on %, and on %r\ ^^ we have Z
{5.11)

A = l

Proof. Since (z, t) e % n ^fc, ({,(z, r), t) e L/^ x A and (4(z, 0, 0 e f/fc x A are the same point on M^, we have ^;(^, 0 =/;]^(4(^, 0, 0,

a = l,...,n.

Since ^^=f]LUk,t) are holomorphic functions of Ck,-^ , CLh,... applying d to both sides of (5.78), we obtain

(5.78) ,t^,

< ( ^ , 0 = Z ::7i-am^,0. Hence, we have ,-^ y

r > L y ..A^^ ^>^^ ;.^^

^ = 1 O'^/c A

On the other hand, applying d^ to both sides of (5.78), we obtain

d.cj{z.t)= i %d,a{z,t). /3 = 1 OL,k

Hence

I cp]{z, t)d,Cj{^^ 0 = 1
A

Therefore, since det(aA^7(^, 0 ) ^ 0 , if we denote the domain of the local coordinates ( z \ , . . , z " ) by (7, we have (pj{z, t) = (pi{z, t) on %r\%Lkr^ [/ X A 7^ 0 , hence X (p;^(z, O^A^ Z
there. Since ( z \ . . . , z") are arbitrary local complex coordinates on M, we obtain (5.77). I

§5.3. Theorem of Existence

263

If for (z, r) G % we define n

From (5.77) (p{t) = (p{z, t) is a C°° vector (0,1)-form on M for every In terms of local complex coordinates we have

viz,0=1

tsA.


A=l

In the sense that
cp'iz, t)d,i,(z, t),

a = 1 , . . . , n.

(5.79)

A =

If we consider (p(t) = (p{z, t) = J^l^^(p\z, t)dx as a differential operator which associates to every local C°° function / ( z ) a local (0, l)-form ^"^1 (p^(z, t)dxf{z)^ (5.79) can be rewritten as (5-(p(r))^;(z,0 = 0,

a = l,--.,n.

(5.80)

Since ^/(z, 0) are holomorphic function of z \ . . . , z", 5^/(z, 0) = 0. Hence by (5.79), we have (p(0) = 0.

(5.81)

Theorem 5.3. If we take a sufficiently small polydisk A, then, for te^, a local C°° function f on M is holomorphic with respect to the complex structure Mt if and only if f satisfies the equation (d-
(5.82)

Proof. W r i t e / = / ( ^ j , . . . , ^j) as a C°° function of the local complex coordinates ^] = Ijiz, t),..., ^ ; = ^;(z, 0 on M, Then id-
t),...,

^;(z, 0 )

= I(a-
Cd-
fx^l

I

Oi,j

264

5. Theorem of Existence

Since ~d^Cf = l^ (p^dxCf, we have

Cd-
V X \

IX

drM-^. J

O^j

Since the determinant

is invariant under change of local coordinates ( z \ . . . , z"), it is a C°° function on M X A. Since
cp^Az, t)cp'^(z, t))

7^0

(5.83)

provided that A is sufficiently small. Therefore, since by (5.75) dQt{dx^") ¥=• 0, (d-(p(t))f= 0 if and only if df/d^" = 0 for a = 1 , . . . , n, which means that f=fUj,..., ^7) is a holomorphic function of the local complex coordinates ^],...,^;onM, I This theorem implies that the deformation Mt of the complex structure on M is represented by the vector (0,1)-form ip{t)= i


A=l

i

i


A = l v = l

on M. (p(t) satisfies the following conditions induced from (5.79). Denote Cf{z, t) by Cf and
A= l

A= l

A= l

Hence A= l

/u, = l

Since 5,^; = 1^ (pUxC" by (5.79), we have _

n

_

_

ddfxf = Z d^d^SJ" dz"' = X d^d^Cf dz"" V—l

V

§5.3. Theorem of Existence

265

Consequently putting

we obtain

IX

A \ /x

/

A ^t

while, since cp^ Mp^ = -(p^ A
A = l \/A = l

A= l

/

Since det(aA^") ^ 0, we obtain dcp'{t)= i cp''(t)Ad^ip'{t), /. = !

(5.84)

Thus 9 ( 0 = l L i
.^ .;^ ,//_r-n^^,//^ A;I m^) 1 (
(5.85)

A = l )a = l

where we put

^

ql

for i/^'^ = (1/^!) X il/i^...i}^(z) dz""^ A • • • A d z \ [<^, i/^] 15 independent of the choice of local complex coordinates ( z \ . . . , z") used for its definition^ and we have [
Therefore, using the bracket, we write (5.84) as ~dcp{t) =

'±
(5.86^

266

5. Theorem of Existence

For the bracket we can easily verify the following formulae: For cp e ^^'^(T), if/e^e'^'^T) and TG^^'^(r), [iA,^] = - ( - i r [ ( P , H

(5.87)

d[cp,ip] = [dcp,ilj] + {-incp,~dH

(5.88)

{-ir[l
(A = 0.

(5.89)

We call

q^l.

(5.90)

For the infinitesimal deformation {dMt/dt)t=o, we have the following theorem. Theorem 5.4. {d(p(t)/dt)t=o is a d-closed (0, l)-/orm, and under the isomorphism /f'(M,e) = ^P(r)/a^^'^(T), {dMt/dt)t=oeH\M,(d)

corresponds to

(5.91)

-(d(p(t)/dt)t=o^^f\T).

Proof. Put

.,= i(2S^)

^.

(5.92)

The infinitesimal deformation (dMt/dt)t=oe H^(M,S) is the cohomology class of the 1-cocycle {djk}eZ^({Uj},&). We fix a tangent vector d/dte ro(A), and for a C°^ function / ( O of ^ G A , denote ((9/(0/^0r=o by / Differentiating (5.78):

in^^t)=ff,u,(z,t),t) with respect to t, and putting r = 0, we obtain 4j - L 77^^fc"^\ ^ = i5ffc

\

T;

/

dt

/ t=o

'

where ^^ = ^/(z, 0) and ^f = ^^(z, 0). Therefore putting

§5.3. Theorem of Existence

267

for each 7, we have ^jk = ^j -

^k-

Since tj = {dCfi^, t)/dt)t=o is a C°° function on Uj, ^j is a C°° vector field on Uj. Next, differentiating (5.79):

A= l

with respect to t, and putting t = 0, we obtain

A= l

since ^^(z, 0) = 0 by (5.81). Hence

^=i:Hr—^=i:i: 6 ^ ^ ( 7 ) ^ 5 ^ 2 ^ ( 7 ) ^ 0 induces the exact sequence of cohomology groups 0-> H\M, 6 ) -> if^'^T) - i ^ P ( T ) ^ f / ^ M , e ) ^ 0, from which the isomorphism / / ^ M , 0 ) = ^ P ( T ) / a i ^ ^ ' ^ ( r ) is induced. Thus for the 0-cochain {|,}G C^(U, si{T)), we have d{^j} = (p, and 6{^,} = -{6|yfc} from the above results. Consequently by the definition of 5*, we have

,*^_(^A

, ,here

^ =(

^

. I

As stated above, a deformation M^ of M is represented by the vector (0, l)-form (p(t) on M satisfying the condition (5.86): d(p{t)=^(p(t), (p{t)]. In order to clarify the meaning of this equality, fix t for a moment, and assume that z moves on the domain (7 of a single local complex coordinate system ( z \ . . . , z"). Wewrite(^(Oas (p =X"_i
X (p^^A,

^= l,...,n,

A= l

the equation (5.82)

( 5 - X A
is written in the form of simultaneous

268

5. Theorem of Existence

partial differential equations as LJ=0,

p = l,,,.,n,

(5.93)

From (5.83) det(5t-I^(;p^(;p^) 5^0. It follows from this that 2n partial differential operators L j , . . . , L„, L i , . . . , L„ are linearly independent. In fact if

A= l

\

/u,

/

/x = l

\

A

/

then a^-=Y^^(p%h^ and h^=Y.^.^^cl^ hence a^=Y.^^^(p^(pla,. Therefore by (5.83) AA = 0 for A = 1 , . . . , n, hence also b^ = 0 for ;u = 1 , . . . , n. (5.86) is written in the explicit form as ^A _ dr(pi-d,Cp',=l(p'^d^
and is equivalent to the system of equations L , L . - L . L , = 0,

T,^ = l , . . . , n .

(5.94)

As is clearly seen from the proof of (5.84), (5.94) is a necessary condition for the system of partial differential equations to have n C°° solutions / = ^« = ^«(z), a = 1 , . . . , n, with dQiid^^J") 9^ 0. For the sufficiency of (5.94), Newlander and Nirenberg proved the following theorem. Theorem 5.5 (Newlander-Nirenberg [26]). Let

T, ^ = 1 , . . . , n.

Then the system of partial differential equations LJ=0,

p=

l,,..,n,

has n linearly independent C°° solutions f= ^" = ^"(z), a = 1 , . . . , n, in a sufficiently small neighbourhood of any point of U. Here the solutions ^^, - -- ,C are said to be linearly independent if J

J. ^\^

?**'55

5r> > • • • ? »

^ /L r\

fc

c\c\

§5.3. Theorem of Existence

269

For solutions / = ^^, a = 1 , . . . , n, of the system of equations L^/= 0, p = l,... ,n,wQ obtain by a simple calculation the following equality:

diz\...,z",z\...,n

(st-X-P^-P^)

det

diz\...,z")

Therefore if the solutions ^ \ . . . , ^" are linearly independent, we have detC^xD^O. Consequently when detC^t-X^
Then, if we take a sufficiently fine finite open covering { Uj) of M, by the Newlander-Nirenberg theorem, there are n linearly independent C°° functions CJ - Cf{z), a = 1 , . . . , n, on each Uj such that (d-i^
(5.96)

Since '*'%(z\...,z",z-...,z-")'^"' ^^: z^ Ij^z) = ( ^ j ( z ) , . . . , ^/(z)) gives complex coordinates on L^ c: M if M is considered as a differentiable manifold. By the proof of Theorem 5.3, a local C°° function / o n (7 is a holomorphic function of ^ ] ( z ) , . . . , ^"(^) if (d-(p(t))f= 0. Therefore on UjnUk^^ 0 , ^7(^) are holomorphic functions of ^i(z),..., ^fc(z), namely, ^;W=/;.(^fc(^)),

a = l,...,n.

Consequently, {^i,..., {y,...} is a system of local complex coordinates on the differentiable manifold M, hence, defines a complex structure on M, which we denote by M^. If (p is equal to (p{t) defined by (5.79), it is clear that M^(,) = Mt = m~\t). With these preparations, we prove the following theorem of existence.

270

5. Theorem of Existence

Theorem 5.6 (Theorem of Existence [18]). Let M be a compact complex manifold and suppose H^{M,S) = 0. Then there exists a complex analytic family {M, B,m) with Oe Be: C"" satisfying the following conditions: (i) (ii)

m-\0) = M. po'. d/dt-^(dMt/dt)t=o ^ith Mt = Tx7~^(t) is an isomorphism ofTo(B) ontoH\M,e): To{B)^ H\M,e).

Before we give its proof, we will explain the motivation for this theorem. According to the conjecture given at the end of the preceding section (a), we expect that m(M) = dim H^(M, 0 ) holds if the number of moduli m(M) of M is defined. If m{M) = dim H^(M, B) holds, by Definition 5.4 of the number of moduli, there is a complex analytic family (M, B,m) satisfying the conditions (i) and (ii) of the above theorem. If there is a complex analytic family satisfying the conditions (i) and (ii), then from the consideration in the preceding section (a), we see that for an arbitrary 6 G H^(M, B ) , all obstructions T2 = j[^, 0], Fs, r 4 , . . . must vanish. Since there exists a compact complex manifold M for which there is a $eH^{M,S) with [$, 6] 9^0, it is absurd to try to prove the existence of a complex analytic family satisfying (i) and (ii) for an arbitrary M. As stated in (a), if //^(M, 0 ) = O, all the obstructions vanish, while we could find no other simple condition for all the obstructions to vanish. So we tried to prove the theorem of existence for M with H^{M, 0 ) = 0. Assume that there is a complex analytic family {M, B,m) satisying (i) and (ii). Take a sufficiently small A with OG Ad 5, and define by (5.79) the C°° vector (0, l)-form
(p^e^f^(T)

,

A = l,...,m,

ot^ / t=o

by Theorem 5.4, and {^i,..., <^m} forms a basis of

Conversely, given p^, G ^f\ T) for A = 1 , . . . , m, such that {jSi,..., jSml forms a basis of H^{M, 0 ) , assume that there is a family {(p{t)}\teA} of C°° vector (0, l)-forms (p{t) on M with 0 G A c: C"", which satisfy the integrability condition d
'i(p(t),(p(t)l

and the initial conditions
( ^ ) \

Ot^

=p„

A = l,...,m.

(5.97)

/ t=Q

Since A is assumed to be sufficiently small, we may assume that (p(t) =

§5.3. Theorem of Existence

271

ZA SV
We shall fix some notation. For a power series P{t) in ti,...,t^, denote by

we

its ^th homogeneous part. Thus oo

p{t)=

i:

PAQ)=Poit)+---+PM+---.

Further we put p ^ ( 0 = Po(0 + - - + P . ( 0 . Given two power series P{t) and Q{t), if P^it) = Q'^it), we denote it by P{t) = Q{t), We must find a power series (p{t) satisfying the integrability condition and the initial conditions. If we put m

^'o^O,


(5.98)

A= l oo v= \

satisfies the initial conditions (5.97). Consequently it remains to determine
(5.99)

272

5. Theorem of Existence

holds. Since l3^e^-/iT), we have d(pi(t) = 0, and [(p^it),(p,{t)] is a homogeneous polynomial of degree /JL + I^ in t^,..., t^. Therefore the equality (5.99) is reduced to the system of infinitely many congruences d
p = 2,3,....

(5.100).

Thus in order to obtain (p(t), it suffices to determine
(5.101)

Putting ^2(t)=^(Pl(t),(Pi{t)l since d(pi{t) = 0, we obtain from (5.88) and (5.87) dip2(t)=[d(p,{t),(pM] Therefore ilf2 = ll^^,ikxM^ assumption

= 0.

with ^AA;. = ^ M ^ ^ ' ' ( r ) .

Since by the

^ F ( T)/d^'''\ T) = H\M, e ) = 0, there is a cp;,^ e ^""'^T) we have

such that

d
IAA/X=5
If we put (piiO =1. (Px^^tJ^,

Ht)=K
Thus (5.110)2 hold. Assume that already (p''(t) = (p^(t)-\-' • • + <;P„(0 is determined such that (5.100), holds, and consider (5.100),+i. Since (p'"^\t) = (p''(t) +
(5.102)

Since [^"(0, •p'CO] = .[«p'"HO, ^ " " ' ( 0 ] , and (5.100), holds by the hypothesis of induction, the right-hand side of (5.102) =„ 0. Let ip^+iit) be the (i' + l)th homogeneous part of the right-hand side of (5.102). Then we have ^ . + i ( 0 .

(5.103)

§5.3. Theorem of Existence

273

Thus the congruence (5.102) is reduced to the equation 5«p,+i(0 = ^.+i(0.

(5.104)

By (5.103), (5.88), and (5.87),

v+l

while dip'^it) =^{l/2)[(p''(t), have

(p'^it)] as stated above. Since cp'^it) =oO. we

while by (5.89) [[
Consequently, since / / ^ ( M , e ) = 0, by (5.90) we can find (p,^...,^e5£'''\T) such that ifjj,^...^^. Thus there is a solution

i^j H

1- Vf^ =

v+\

of the equation (5.104): ~d(p,+M = dz^®dz~'' on M, and define dual forms, inner products, norms, etc. of differential forms on M as in §3.5(a). For the tangent bundle T, we define its Hermitian metric on fibres ^y ^",^^=1 Sxd^r for XA ^^^A e T, Then the inner product of vector (0, q)forms

274

5. Theorem of Existence

We denote by b the adjoint operator of d with respect to this inner product. Thus Also we put

n = ab+ba. (pe^^'^{T) is called a harmonic vector {O,q)-form if d(p=t>(p = 0. (p = if°'^(r) is harmonic if and only if Hip =0. The vector space

consists of all harmonic vector (0, g)-forms on M. By Theorem 3.19, we have ifO'^(r) = H^'^(T)©nif''''^(T).

(5.105)

From this it follows that

^f'^(T) = H''^(r)ea^^'^-nT).

(5.106)

H'^iM, e ) = ^f'^(T)/dL'''^~\ T) = H^'''( T).

(5.107)

Hence

Next we define the Holder norm. First l e t / ( x ) = / ( z \ . . . , x^") be a complex-valued C°° function defined on a domain U of IR^". We denote any partial differential operator of order of order h with respect to x \ . . . , x^" by D'':

Let /c be a non-negative integer, and a a real number with 0 < a < 1. Then the Holder norm |/|J^+« o f / = / ( x ) is defined by \f\k+a= I

I sup|D/(x)|+X

h=0 j^h xeU

sup

jyk x,y&U

— l^^ ~

, J]

where \x-y\=yJY,^\x''-y'^f', and the summation YJD^ is taken over all partial differential operator D^ of order h. For a C°° vector (0,q)-foTm (pe^^-'^iT) on M, we define its Holder norm \(p\k+a as follows: Let {Uj} be a finite open covering of M, where each Uj is a coordinate polydisk. Then by the definition of coordinate polydisks, there are local complex coordinates {Zj} with Zj = {zj,..., z") satisfying the following conditions: (i) (ii)

[ Uj] is contained in the domain of z,. ^ = {z,||z]|
§5.3. Theorem of Existence

275

Putting zj = x]"""^ + ixj"", we introduce the real coordinates Xj = ( x ] , . . . , xj"). If we represent cp = if^'^(T) on each Uj in the form Z - | E Mj^ dzj^ A • • • A dzj^ ^ , ^%,-i>q =
L^GR^".

Define

|
the inequality

|^U+«^c(p(^U-2e«+klo)

(5.108)

holds for (p e if^'^(T), where c is a constant depending only on k and a. We give proof of this theorem in §4(b) of the Appendix. The fundamental inequality (5.108) is called the a priori estimate. With these preparations, we consider if^'^(T) for M with H^(M, 0 ) = 0. In this case since H^'^( T) = 0 by (5.107), if D lA = 0 for (^ e ^^'^( T), we have (/^ = 0, and by (5.105)

^^''(r) = n=^^''(T). Therefore the linear map D maps if^'^(T) in a one-to-one manner onto itself. We denote its inverse by G: G = n ~ \ G is called the Green operator. It is clear that (A = nG(A,

il/e£e'''\T).

(5.109)

iA = abG(A,

ipe^f\T).

(5.110)

If dif/ = 0, we have Proof. We have il/ = OGil/ = dhGil/ + 'i>dGil/. Applying d to this equality, we obtain dhdGil/ = dil/ = 0. Therefore ||baG(Af = (hdGil/, bdGil/) = (dbdGil/, dGil/) = 0. Thus 'i^dGifj = 0. Hence ij/ = dhGilj.

276

5. Theorem of Existence

Lemma 5.7. For /c ^ 2, the inequality |G(AU+a^C,|(AU-2+a

(5.111)

holds for ijjG if^'^(T), where Cj is a constant independent of ijj. Proof Since DGil/ = ij/, by the a priori estimate (5.108) we have |G(AU+.^c(|iAU_2+« + |G(A|o)

for il^e^''\T),

(5.112)

Therefore in order to prove (5.111) it suffices to show the inequaUty \Gilf\o^C2\ilf\k-i^^

for ilje^''\T),

(5.113)

where C2 is a constant independent of ij/. Suppose that (5.113) does not hold for any C2. Then for any natural number m, there exists il/^""^ e ^^'^(T) such that |G(A^->|o=l,

|(A^'"1.-2-.«<-. m

(5.114)

Putting (p^"^^ = Gil/^'^\ we have from (5.112)

Thus, representing (p^"^^ on each Uj in the form \ = l ^ I3,y = l

oZj

we have

Fix jf, A, /3, and y for a while, and p u t / „ =fm(Xj) =
|Dt/„(x,) - D';uyj)\ s 2c|x, - y,r. and |DJ'/^(x,)-DJ'/^(y,)N2c|x,-j,|

for/z
Therefore for each Dj,h^k, the sequence of functions {/^^/^(x,)} is equicontinuous in U,. Clearly \D^f^{Xj)\^2c on U^ for /i ^ /c. Consequently, by Ascoli's theorem, we can choose a subsequence {fm(u)} of {/„} with

§5.3. Theorem of Existence

277

m(l) < m(2) < • • • < m(^) < • • • such that the sequence {Djfm(a,)(Xj)} converges uniformly on Uj for each h^k. Writing {/m(i),/m(2), • • •} simply as {/ml = {/i,/2, • • •}, we may assume that {Dffm(Xj)}. converges uniformly on Uj for any Dj with h^k. Put/(x^) = lim^^oo/m(^j)- Then/(x^) is a C"^ function on Uj, and {Djf^(Xj)} converges uniformly to Djf(Xj) for any Dj with h^k. Since the above results hold for each quadruple of indices j , A, ^, y, replacing {JA^'"^} by an appropriate subsequence, we may assume that there is a C^ vector (0, 2)-form cp such that for any Dj with h^k, the sequence {D^cp^j;^^\xj)} converges uniformly to D^(p)^y{Xj) on Uj where ^^"'^ = GiA^'"^ Therefore cp^'"^ converges to (p uniformly on M if m-»oo, and, since / c ^ 2 , Dip^"^^ also to D ^ uniformly. Since cp^"^^ = Gi^^"^ and n(p^'"^ = DGilj^'^\ we have |G^(-)-^|o-^0

(m->oo),

Therefore, since by (5.114) IrA^'^lo^ l'A^'"1k-2+a ^ 0 (m ^ 0), we have n

{cp,ilf) = icp,nGiij) = ia(p,Giif) = o, which implies that

0. I Suppose given a power series in t^,..., t^ oo

pit)=

I p.,...„^r'• • • C-,

n,—„ec.

A power series oo

is said to be a majorant of P ( 0 if |P^^...^^| ^ a^,...,^,

z^i,..., ^^ = 0 , 1 , . . . ,

and is denoted by P(t)«ait). For a power series oo

278

5. Theorem of Existence

we define |(/f|fc+a(0 by OO

vi,...,Pffj=0

We write

if i
^ 1 , . . . , ^^ = 0 , 1 , 2 , . . . .

In order to prove the convergence of the power series <^(0 = ZT=i 0, with respect to the Holder norm, it suffices to show that there is a power series a(t) which converges absolutely in Ag such that \(p\k+^(t)«ait). Since in the construction of (p{t) = 'Z°^^i
Since dil/^+i{t) = 0, if we put

^xh

A= l

first, and define G[
...

A{t)=—-

b X- rih^-'-^u'^ 2 —.

16c v=i

1^

(5.115)

§5.3. Theorem of Existence

279

with ^ > 0 and c > 0. ^ and c will be determined later. As regards A{t), we have the following inequality A{tf«-A{t), c

(5.116)

Proof. Consider the power series ^(5) = XlLi s""/ v^ in the variable s. Then 00

while, since fx^v/2if\-\-/ji

o'^ 00

00

«/^

1

= p and A ^ ^t, we have

Z.22 — ^ L v2 2 ^ ^ Z- . 2 2 A+ya-i/ A /Lt \+fjL = i,A fl A=l A P

2 Z. x2V A= lA

As is well known, Xr=i 1/A^= 7rV6<2. Therefore 00

^ -

B ( 5 r « 1 6 Z —=16B(5). v=2 V

Since A ( 0 = (^/16c)B(c(^i + - • • + r^)), (5.116) follows immediately from this inequality. I Fix a natural numberfc^ 2. We will show by induction on v that if we choose appropriate large h and c, k1..«(0«A(0

(5.117).

holds. For ^ = 1, (^^(r) = /3iri + - • '-^Pmtm, and the linear term of A ( 0 is (6/16) (^14- • • • +1^). Therefore, if we choose b such that |j8A|fc+a < 6/16 for A = 1 , . . . , m, (5.117)i: \(p\^M« A{t) holds. Now we assume that (5.117). holds, and make an estimate of |(^.+i|fc+a(0By (5.115)

By the definition of the Holder norm, we have the inequalities |b(AU+«^i:i|eAU+i+«,

il^e^''\T),

(5.118)

and \[cp, (A]U-i+« ^ K2l
(^, tA G ^''\T)

(5.119)

280

5. Theorem of Existence

where K^, K2 are constants independent of if/ and (p. By (5.111) and (5.118),

hence |<^.+lU+a(0« J^lCl|lA.+ lU-l + «(0. Since (A.+i(0=l[9'(0,
Thus (5.117)^+1 holds, which completes the induction. Therefore we have showed that \
(5.120)

Since the radius of convergence of the power series ^^=1 ^""/^^ is equal to 1, A{t) converges absolutely for teA^ if 0 < e ^ l / m c . Hence by (5.120) ^ ( 0 = Z ^ = i ^ ^ ( 0 converges with respect to the Holder norm | 1^+^ for r e Ae. Consequently (p{t) is a C^ vector (0, l)-form on M x A^, namely, if we write (p{t) on each Uj x A^ in the form 'Pit)=l\l
OZj

the coefficients (pfpy{Zj, t) are C^ functions of z^,.. •, Zj, ^1, • • •, tm- Although k is an arbitrary natural number not less than 2, we cannot deduce immediately from this result that (p{t) is C°°, because the constant c = K^K2Cxh above does depend on /c, hence it does not follow immediately that there

§5.3. Theorem of Existence

281

is an e > 0 such that e ^ 1/mc for all k. We must prove C°° differentiability of (p{t) in another way. By (5.107) we may consider that {j8i,..., ^8^} above is a basis of H^'^T). Then for 0 ^^""^A satisfying the integrability condition 5
282

5. Theorem of Existence

with
dz''

By 5
If we put

dZ^

A=i

dZ

the partial differential equation {d — (p)f=0 is reduced to the system of partial differential equations: W=0,

^ = l,...,n,

—/=0,

)u. = l , . . . , m .

(5.122)

L i , . . . , L„, L i , . . . , L„, d/dt,..., d/dtm, d/dt^,..., 5/^^^ are linearly independent. Consequently, by the Newlander-Nirenberg theorem, (p defines a complex structure M on M x Ag. If we choose a sufficiently fine locally finite open covering {L^} of M, and take a sufficiently small A^, the equation (5.122) has n-\-m linearly independent solutions f=(f(z,t), p= 1 , . . . , m + 71 on each Uj x A^, and the map

gives local complex coordinates of M on Uj x A^. Since f=tf^ is clearly a solution of (5.122), we may assume that (J'^^{z, t) = t^j, for )LC = 1 , . . . , m. Then we have ^j:{z,t)^U]{z,t),,.,,^^(z,t),t„,..,tJ. Therefore m: U]{z, 0 , . . . , C(^. 0 , ^ 1 , . . . , U -> (tu . . . , ^m) is a holomorphic map of M onto A^. For each ^G A^, tD'~^(0 is a complex manifold whose system of local complex coordinates is given by

§5.3. Theorem of Existence

283

{(^](z, 0 , . . •, C]{z, t)}. Since/=_^/(z, 0 , jS = 1 , . . . , n, are linearly independent solutions of the equation {d-(p{t))f=0 on U^, m~^{t) = M^i^^y Thus {M^(,) 11 e Ag} forms a complex analytic family (Jt, A^, m). Thus Theorem 5.6 (Theorem of Existence, [18]) is proved. Since each component ^/(z, t) of the local complex coordinates (f](z, t),..., ^/(z, 0 ) of M^(f) is a solution of (5.122), a^f(z, O/^f/it =Ofor ^t = 1 , . . . , m. Therefore C°° functions ff (z, 0 of z \ . . . , z", ^ i , . . . , ^m ^^^ holomorphic in ^ i , . . . , t^^. About the same time with us, Grauert proved this theorem of existence by an elementary method. He explained his remarkable idea at "Nothing Seminar" under the direction of D. C. Spencer. The main point of his idea is in replacing the fundamental equalities (5.62):

m^k,t)^frj{Mz^,t),t) by the inequalities

\mz,,t)-frj(fj,(zj,,t),t)\<8. By this method he succeeded in proving the convergence. Later he proved the most general theorem of existence of deformations for complex analytic spaces based on this idea ([9]). Later another proof of Theorem 5.6 by an elementary method was given by Forster and Knorr ([5]). We note that the theorem of completeness (p. 230) was not known at that stage of the development of the theory of deformations. However, since the proof of the theorem of completeness is elementary and independent of the existence theorem which is the main theme of this chapter, in verifying the equality m = m(M) for concrete examples of M (pp. 230-248), we replaced for simplicity our original argument ([20], §14) by one using the theorem of completeness.

Chapter 6

Theorem of Completeness

§6.1. Theorem of Completeness In this section we shall prove the theorem of completeness stated in §5.2(b). Let {M, B, TJT) be a complex analytic family, B a domain of C" containing 0, and pt'. Tt(B)^ H\Mt,St), M = ^ HO, the linear map defined by (4.22): d

( d\

dMt

Theorem 6.1 (Theorem of Completeness, [20]). If po'. To(B)->H^(Mo, Bo) is surjective, the complex analytic family (Jt,B,m) is complete at Oe B. Recall that we mean by the completeness of (M, B, m) at 0 that given any complex analytic family (J^, D, TT) such that Oe D^C^ and 7r~\0) = MQ, if we take a sufficiently small subdomain A with 0 G A <=: D, and let ^ A = 7r~^(A), we can find a holomorphic map h: s^ t = h{s), h(0) = 0, of A into B such that (Jf^, A, TT) is the complex analytic family induced from (Jt, B, m) by h (see p. 228, Definition 5.2). If {J^A, A, TT) is the complex analytic family induced from {M, B, tzr) by h, then for each 5 G A, Ns = rr'^s) = Mh(s) x s, and

is a submanifold of ^ x A (see p. 206). Let g be the restriction of the projection JixA-^Ji to Jf^- Then g is obviously a holomorphic map of J{A into M. Moreover g maps each N^ biholomorphically onto Mh(s)- For, writing a point of Ns = Mh(s)^s as {p,s), ; ? G M ^ ( ^ ) , we have g{p,s)=p. Identifying (p, 0) G No = MQ xO with p e MQ, we may consider 7r~^(0) = No = MQ. Then, if we denote the identity map {p,0)->p by go, g: J^^-^Ji is an extension of go'. NQ-^ MQ. Conversely, the following lemma holds. Lemma 6.1. If we can extend the identity go'. No = Mo ^ Mo to a holomorphic map g\ Jf^-^ M such that g maps each N^, 5 G A, biholomorphically onto Mh(s), then (Jf^, ^, ^) is the complex analytic family induced from {Jt, B, m) by h.

§6.1. Theorem of Completeness

285

Proof. Let (JV, A, TT) be the complex analytic family induced from (M, B, m) by h: A^ B. Then for each 5G A, 7r'\s) = Mh(s)^s and

is a submanifold of J^ x A. Denoting a point of Jf^ by ^, we consider the holomorphic map ^: q^^{q) = {g(q), 7r(q)) of ^ A into MxA. Since g maps each N^ biholomorphically onto M^(^), O maps N^ biholomorphically onto

Hence O maps Moreover

^A^U^GA^^

biholomorphically onto ^ = U5eA ^^(^) ^'^•

7f
M = Uj%.

%=

{{Cj,t)eC-xB%\<\},

and m{^j, t) = t Of course Uj is a finite union. If UjnUk9^0, (^fo i) are the same point of M if Cj = gjkUk, t) = {g]k{Cu. 0 , . . . , gUCu. 0 ) ,

Uj, t) and

(6.1)

where the g%{Ck, 0, OL = \,. .. ,n, are holomorphic functions on Uj n U^.

286

6. Theorem of Completeness

Similarly we assume that D = {5 E C' | |s| < 1}, J^ = UjWj,

^ , = {(Z,,5)GC"XD||Z,|<1},

7r{zj,s) = s, and that for WjnW^y^ 0 , (z,, s) and {zj,, s) are the same point of Jf if Zj =fjk{zk, s) = {fjkizk, s),.,.

Jjkizk, s)).

(6.2)

Moreover, since by hypothesis, NQ = MQ, we assume Wj n No = %nMo and that the local coordinates ({,, 0) and {zj, 0) coincide on Wj n No= % n Mo, namely, if ^] = z ] , . . . , ^J = z/, ({y, 0) and (z^, 0) are the same point of WjnNo= % n Mo. Putting bjjc(zj,)=fjdzk,0),

(6.3)

from (6.1) and (6.2), we have bjkUk) = gjMk,0).

(6.4)

Thus we have No = Mo = U ^ - ,

Uj=WjnNo=%nMo,

(6.5)

j

and {zy}, z^ = (z],..., zJ), is a system of local complex coordinates of the complex manifold No = Mo with respect to { L^}. The coordinate transformation on Uj n Uk is given by zf = bf,iz,),

a = l,...,n.

Our purpose is to define a holomorphic map h: s^t = h{s) with h{0) = 0 of Ag = {5 G D11^1 < E} into B for a sufficiently small e > 0, and at the same time to extend the identity go'. NQ^ Mo= NQ to a holomorphic map g: 7r~^{A^)^M such that m ° g = h° TT. The condition m ° g = h^ IT means that the holomorphic map g maps each Ns = 7r~\s), ^eAg, into M^(5) = m~\h(s)). But then, since g is an extension of the identity go, it is easy to see that, if we take a sufficiently small 8<8, g maps N^ biholomorphically onto Mh(s) for l^j < 8. Replacing e by such 5, we may assume that g maps Ns biholomorphically onto Mh(s) for s e A^. In what follows, we write A = A^ and ^ A = '7'" H^) for simplicity. Suppose for the moment that we can construct such h and g. Then since giUj)= UjCi %ij for each Uj^ No, g~^(%)^«^A is an open set containing U;. Therefore we can choose a continuous function e{Zj) of z, on Uj with 0 < 8(z,) ^ e such that the subdomain Wf =

{{Zj,s)\\Zj\<\,\s\<e{Zj)}

§6.1. Theorem of Completeness

of Wj is contained in

287

g~\%): UjCzWfczg-\qij)nWj.

Since the holomorphic map g maps Wf into %, g can be written on Wf in the form g: {zj, s) -> Uj, t) = (gjizj, 5), his)),

(zj, s) e Wf,

(6.6)

where each component gf(zj, s) of gj(zj, s) = {g](zj, 5 ) , . . . , gj{zj, s)) is a holomorphic function of z],..., z", ^ i , . . . , Si defined on Wf. Since g:izj,0)-^Uj,0)

= {zj,0)

gj(zj,0) = zj,

h{0) = 0.

is the identity, we have (6.7)

Expanding gj{zj, s) into power series of 5 i , . . . , Sj, we write it in the form 00

gjizj, s) = 2,- + X gjiAzj, s),

(6.8)

where, in the notation of § 5.3(b), gji^zj, s)=

X t'jH

g>,.--/(%)'^r' • • • ^?

\-vi=v

is a homogeneous polynomial of degree ^' of 5 i , . . . , 5/, and each component S^vyvii^j), « = 1 , . . . , n, of the coefficient

is a holomorphic function of z],..., z" defined on Uj. For simplicity we express such situation as gj^j...^^(zj) is a vector-valued holomorphic function defined on Uj. In general the region of convergence of the power series gj(Zj, s) of 5 i , . . . , Si may depend on z^, but they all converge for \s\ < s{zj). Expanding h(s) into power series, we put CX)

Hs)=^ I K(s).

(6.9)

For Wf nWt9^0, if (z,-, 5) G Wf and (z^, 5) G Wf are the same point, then their images by g, namely, Up t) = (gj{Zj, s),his)) and Uk,t) = (gki^k, s), h(s)) are, of course, the same point. Hence by (6.2) and (6.1), if Zj =fjk(Zk, s), then we have gj(zj, s) = gjkigki^k, s), h(s)). In other words, for

288

6. Theorem of Completeness

(zfc, s)eWtn

Wf, the equality gjifjkizj,, s), s) = gjk(gk(z},, s), h{s))

(6.10)

holds. If we expand both sides of this equality into power series of Si,..., Si, their coefficients are holomorphic functions on Uj n U^. Hence, from (6.10) we obtain equalities between these holomorphic functions defined on Uj n L4. We note that, if oo

gj(Zj,s)

oo

= Zj-^ X gjUZk,s),

gk(Zk,s)

= Zj,+ X

gk\AZk,s),

oo

and

h(s)= X! K(s)

are all considered to be formal power series of Si,..., still makes sense. For, by writing down (6.10) as oo

fjk(z,s)-\-

/

S gj\uifjkizk,s),s)

= gjJzk-\-

Si, the equality (6.10)

oo

oo

X gkU^k^s),

I

\

h,{s)),

we can obviously see that, for each /x = 1, 2, 3 , . . . , the terms of degree at most /x of the power series expansions of both sides of (6.10) contains no gj\X , ^), gk\v{ , 5), K{s) with v> yu. Thus assuming that we obtain holomorphic maps h and g with the required properties, and representing g on each domain WJ in the form (6.6): {zj, s)^(gj(zj, s), h(s)), we have seen that the quality (6.10) holds on each UJ nU^y^Z). In order to obtain such h and g, we begin with constructing formal power series h{s), as well as formal power series gj{zj, s), for each Uj, whose coefficients are vector-valued holomorphic functions on Uj such that the equality (6.10) holds. In order to avoid confusion, we use a, j8,... to denote the indices of coordinates and use fju, P, ... to denote the degrees. Furthermore, we use the notation of §5.3(b) for power series of ^ j , . . . , Si. In particular, we write h''{s) = h,{s)-^"' gJ{s)

+ h,is),

= Zj-hgjl^(Zj,s)-\-'

'

'-\-gj\,(Zj,s).

The equality (6.10) is equivalent to the following system of the infinitely many congruences: gjifjki^k, s), s) = gjkiglizk, s), h'^is)),

2^ = 0, 1 , 2 , . . . , (6.11),

§6.1. Theorem of Completeness

289

where we indicate by =,, that the power series expansions of both sides of (6.11)^ coincide up to the term of degree K (6.1 1)O means fjk{^k,^) = gjki^k, 0), which clearly holds by (6.3) and (6.4). Now we want to construct h'^is), gJ(Zj, s) by induction on v so that (6.11)^ hold on each Ujn 11^9^ 0 . Suppose that h'^'^s) and gj~\zj, s) are already constructed in such a manner that, for each Ujn U^^ 0 , gJ-'iM^k,

^), s) - gjk{gr\zj,

s), A-'(5))

(6.11)._,

v— l

holds. Since gj(zj, s) = gj~\zj, s) + gj\^(Zj, s), the left-hand side of (6.11)^ becomes gj(fjk(zk, s), s) = gj~\fjk(zk,

s), s)-\-gj\,ifjkizk, s), s),

where gj\i,{Zj, s) is a homogeneous polynomial of degree P of Si,..., ffkizjc, 0) = bjk(zk) by (6.3). Therefore we have

Si, and

gj\u(fjk(zk, s), s) = gj\,(bjk(zk), s), hence, putting Zj = bjk(zk), we obtain gjifjkizk, s), s) = gj~\fjkizk,

s), s)-\-gj\,(zj, s).

(6.12)

The right-hand side of (6.11)^ is given by gjkigKzk, s), h'^is)) == gjk{gr\zk,

s)-^gk\^.{zk, s), h^'-^s)-^

K(s)).

By expanding gjkUk-^^, t-\-u) into power series of ^^ . . . , ^", Wi,..., u^, we obtain gjkUk + ^,t + u) = gjkUk,t)^

I.

0f(^.-1). ^Uk,t)'e^^

13 = 1 oik

r=l

Otr

where . . . denotes the terms of degree ^ 2 in ^ ^ , . . . , f", Ui,... ^u^. Letting K{s) = {hx\Xs),...,h^\^{s)), and writing gr\s), gk\v(s) instead of gr\zk.s), gk\Azk,s), respectively, we have gjkiglizk, s), h^{s))-gjk{gr\zk, -

i

s),

h^-\s))

^^igr\s),h-\s))'g'kUs)

+1 r=\ otr

^-^(gr\s\h-\s))'K^As)

290

6. Theorem of Completeness

+ i^'(gr'(o),/i'-(o))-M*) = Z 7d(^fc,0)-g^|,(5)+I (-^^-T;^—

• ^H-(^)-

Therefore, since g/kCz^, 0) = bjj^izk) = Zj by (6.14), we have

(3^1 OZj^

r=l\

dtr

/ t=0

From this and (6.12), it follows that the congruence (6.11)^ is equivalent to the following:

V p dZ%-

r \

Otr

/ t=0

By the hypothesis of induction, the left-hand side of this congruence = ^-i 0. Hence, if we let F^fci^ denote the sum of the terms of degree v of the left-hand side, we obtain TMAZJ, S)

= gr\Uzu.

s\ s) -gj^{gr\z^,

s\ h-\s)).

(6.13)

Hence (6.11)^ is equivalent to the following: ^jk\v{Zj,

S)

" dz+ I T^'gk\v{zk.s)-gj\^.{Zj,s),

(6.14),

^-1dZ^

where z^ and Zj = bj^iz^) are the local coordinates of the same point of NQ. In order to clarify the meaning of these equations, we introduce holomorphic vector fields as follows:

§6.1. Theorem of Completeness

a^l

291

\

Olr

a=l

/ t = 0OZj

dZj

gk\As)= I

gkUzk,s)—j.

Then in these terms (6.14)^ is written in the form m

^jk\As)=I,

K\As)Orjk-^gk\As)-gj\,(s).

(6.15)^

r=l

By (6.5), U = {Uj} is Si finite covering of MQ = NQ. Since we assume that ^" = z/, a = 1 , . . . , n, the 1-cocycle {Orjk}e Z ^ U , 0 ) represents the infinitesimal deformation dr = po(d/dt,) e H\MO, ©O)- Since the coefficients r^fci^i- -^z of the homogeneous polynomial

^jk\v{s)=

Z

r^-fc.,.-^;-5^ • • • ^p

are holomorphic vector fields on L^ n U^, .

{r,.|.(5)}=

I

{r,fc.,...,}sr'• • • 5,-'

is a homogeneous polynomial of degree v whose coefficients are 1-cochains {r,,,,...,,}€C'(U,0o). Similarly {gj\As)}=

I

{gj.,-.,}S-['• • • s?

is a homogeneous polynomial of degree P whose coefficients are 0-cochains {^jVi •i.Je C^(U, 0o)- Under these circumstances, the equation (6.15)^ is written in the form m

{rMs)}=l

K\As){erjk}+s{gjiAs)}.

(6.i6)„

r=l

Thus in order to construct h''{s) = h''~\s)-\-h^(s), gj(zj,s) = gj~\zj, s) + gjij,{Zj, s) so that (6.11)^ hold, it suffices to obtain solutions hr\As), r = 1 , . . . , m, {gj\As)} of the equations (6.16)^. If solutions K\j,{s), r = 1 , . . . , m, {gj\u{s)} exist, then, since the right-hand side of (6.16)^ is a 1-cocycle on U, {F^fci^C^)} is also a 1-cocycle, i.e., for

292

6. Theorem of Completeness

each Ui n Ujn 11^9^0, we have Tj^lAs)-T,k\As)

+ r,j^As) = 0.

(6.17)

Conversely, if {Tji^\^(s)} forms a l-cocycle, then (6,16)^ has solutions hr\r,is), r = 1 , . . . , m, {gj\As)}. Proof Let

I'jH

\-Vi=v

Then (6.16)^ can be written as m r=l

Therefore it suffices to prove that any l-cocycle {r^fc}GZ^(U, ©o) can be written in the form m

{Tjk} = I K{erj,,} + S{gj},

K e C, {gj} e C°(H, 0o).

(6.18)

Let yeH\Mo,So) be the cohomology class of the l-cocycle {Tjj^}. Since by hypothesis, po* To(B) -» H\MQ, OQ) is surjective, y is written in the form of a linear combination of the Or as m r=l

Consequently the cohomology class of the l-cocycle {Tjk}-{L7=\ K^rjk) on U is 0. On the other hand, by Theorem 3.4 (p. 121), n ^ : H\Vi, Bo) = Z^(U, (do)/8C\U, ©o) -> H\M^,

BO)

is injective. Therefore {r^7c}-Zr=i Ki^rjk}^ 5C^(U, Bo), hence {Tj^} can be written in the form (6.18). I Next we shall prove that {r^7c|^('^)} is a l-cocycle, i.e., (6.17) holds. Proof By representing each term of (6.17) as vector-valued holomorphic functions, we get " dzTik^^{zi,s) = Tij\^{Zi,s)+ Z -r^^fk\v{Zj,s), )3-l

Zi = bij(zj).

OZj

Here Zj and z, = bij(zj) are the same point of MQ. Using matrix notation, we

§6.1. Theorem of Completeness

293

write this in the form ^ikiA^h s) = Tiji,{Zi, s)-^Bij{zj)Tjk\Azj, s),

(6.19)

where ByCz^) = (azf/azf )«,^^i_„. Writing ^('5) for/^(^fc, ^), and gr\s) for gk~\zk, s) for simplicity, we obtain from (6.13), r,-,|.(z, s) - gr\Ms), s)-g,^{gr\s), h-\s)). Since on U^nUinUj^^0,

(6.20)

g,fc(4, t) = gyiUk, t), t), and since by (6.13)

gjk(gr\s), /i->(s)) - gr\Ms), s)-rj,\Azj, s), the second term of the right-hand side of (6.20) is written down as

gikigr\s),h''-\s))=gy(gj^(gr\s),h-'-\s)),h-'-'{s)) = gij(gr'(fA'^), *)-r,fc|.(z,-, s), h'-'is)) V

- i

^igr'(fjkis),s),h-\s))rf,i.(zj,s).

But, since gj \fjk{Zk, 0), 0) = bjt,{zk) = Zj, we have

^ ( g r H ^ ( O ) , 0), 0) = ^ ( z „ 0) = ^ = ^ , dCf ' dCf dzf dzf' hence, we have

gi,.{gr\s), h''-\s)) - g,(gr'(&(*), s), h-'is)) V

" bz-

- 1 -ri^fkUzj, s), p^ldZj

Therefore by (6.20) we obtain

r,.|.(z„ 5) = gr\fikis\ s)-g,(gr\fjk(s), s), h-\s)) -{-Bijizj)Tjk\^{Zj,s).

294

6. Theorem of Completeness

Thus, to prove (6.19) it suffices to verify

r,|.(z, s) = gr\fikis),

s)-g,(gj-\fj^(s),

s), h-\s)).

(6.21)

By (6.13) we get

r,|.(6,(z,), s) = gr\fij{zj, s), 5) -g,(g;-^(z,, 5), h-\s)). Substituting Zj =fjk{zj„ s) and noting bij(fjk(zj,, 0)) = bij{bjk(z,,)) = bij(zj) = Zi, we have ^ijWibijifjkizk, s)), s) = Tiji^ibijifjkizk, 0)), s) ^Tij\^{Zi,s).

In view of the equality ftjifjkizk, s)) =fik(zk, s), (6.21) holds.

I

Therefore {Tjk\i.{s)} is a 1-cocycle, hence, by solving (6.16)^, we can find hr\As), r = 1 , . . . , m, {gj\t,(s)} such that h'^is) and g'^izj, s) satisfy the congruences (6.11)^. In this way, we determine h'^is) and gjizj, s) successively for ^' = 1, 2, 3 , . . . , and obtain the formal power series h{s) and g/(^> s), which obviously satisfy (6.10). (b) Proof of Convergence The power series h{s), gj(Zj, s) constructed in the previous subsection depend on the choice of solutions hr\t,(s), r = 1 , . . . , m, {gj\j,{s)} of (6.16)^,. In general the equations (6.16)^ have infinitely many solutions. In this subsection we prove that, if we choose appropriate solutions K\j,(s) and {gj\j,{zj, s)} of (6.16)^ in each step of the above construction, then h{s) and gj{zj, s) converge absolutely in 1^1 < £ provided e > 0 is sufficiently small. The equations (6.16)^ are reduced to (6.18). We first prove a lemma concerning the "magnitude" of the solutions hi,... ,hm, {gj} of the equation (6.18): m

{r,J= I

K{e^,}+8{gj}.

r=l

We regard holomorphic vector fields F^fc = E" = i r7fc(zy) a/^z", Orjk = Z l ^ i ^^jk(zj) d/dz]", and gj = Y,''^ = i gf(zj) d/dz]" as vector-valued holomorphic functions Tjk{zj) = (rjfc(z^),..., r;fc(z,)), O^jkizj) = (Oljkizj),..., O'^jkiZj)), and gj(Zj) = (gjizj),..., gJ(Zj)), respectively. Then (6.18) is written in the form m

r,fc(z,)=I Mr,7c(z/) + S,fc(zfc)g.(z,)-g,(z,.).

(6.22)

§6.1. Theorem of Completeness

295

Since each L^ = {z^ G C" | jz^l < 1} is a coordinate poly disk, we may assume that the coordinate function Zj is defined on a domain of MQ containing [L/y] (see p. 33). Hence each component dz^/dz^ of the matrix Bjj,(zj^) = (dz]'/dz^)^^p=i^_^n is bounded on L^n Uk9^0. We define the norm of the matrix Bj^izt) by \Bjk{zj)\ = sup

—

= max Z

dzj

feC",

dzi

^#0.

(6.23)

Then there exists a constant Ki such that for all Ujn 11^ = 0 |B,,(z,)|<X„

z.eU^nUj.

(6.24)

Similarly, since the ^^^(z,) = (agj^z^, 0/3'r)(=o are also bounded on L/yn t 4 5^ 0, there exists a constant K2 such that |e.^fc(z;)|
(6.25)

We denote a 1-cocycle {Fj^} by F, and define its norm by |r| = max j,k

sup

|r^fc(z,)|.

(6.26)

zjeUjr\U]^

Lemma 6.2. There exist solutions hr, r = 1 , . . . , m, {gj} of (6.18) which satisfy \K\^K,\Tl

\gj(zj)\^K,\Tl

(6.27)

where X3 is a constant independent of T = {Tjk}> Proof Let 5 > 0, and put, for each Uj = {zj e C" \\zj\ < 1}, L^; = {z,.G^.||z,.|
(6.28)

j

for a sufficiently small 8. We consider solutions /i„ r = 1 , . . . , m, {g,} of (6.18) for a 1-cocycle r = {r^fc} with |r| < +00. it can be easily checked that, on each Uj, \gj(Zj)\ is bounded. In fact, since gj{Zj) is holomorphic on Uj, it is obviously bounded on Uf. If Zj^ Uf, Zj is contained in some L^^ with kr^j, that is, Zj = bjk(zk), ^fcG t/fc. Hence by (6.22) we have m

gjizj)=l

hAA^j) + S,fc(zfc)gfc(zfc) - r,.fc(z,.).

(6.29)

296

6. Theorem of Completeness

By (6.24) \Bjj,{z^)\
^(n^Kiri holds. Suppose there is no such constant K. Then, for each natural number V, we can find a 1-cocycle T^"^ such that t((r^"^) > ^|r^"^|. Replacing V''^ by p(-)/^(p(W)^ we get a sequence of l-cocycles 1^""^ = {T^^}e Z\n, ©o) such that ,(r^-)) = i,

ir^"^l<-.

(6.30)

The equality 6(r^"^) = 1 impHes that there exist solutions /i^/^ {g^^} of (6.18) for r = r^''^ satisfying |/z^/>|<2,

\gy\zj)\<2.

Since, for each Uj, {gy\zj)} = {gj^\zj), g^^\zj),...} are uniformly bounded on Uj, we can choose a subsequence {gj'''^\zj)}, vi < P2< ' ' ' < ^m < • --, ^o as to converge uniformly on each compact subset of Uj. Hence we may assume, from the beginning, that {g-''(z,)} converges uniformly on each compact subset of Uj. Similarly, all the sequences {/Jr*^^}, ^ — 1, 2 , . . . , r, may be assumed to converge. Then we can deduce that each {gy\zj)} converges uniformly on the whole Uj. To show this, we note that Uj is covered by Uf and a finite number of Ui, kv^j, and that, since [Uf]^ Uj is compact, {gj''\zj)} converges uniformly on Uf. For a point ZJE Ujn Ui, i.e., for Zj = bjk(Zk), Zk e Ul, we have by (6.29) m

g]'^\zj)= I hi''^e,j,{Zj) +

Bj,(z,)g'j:\z,)-Ty,\zj).

Since {hi''^} converges and {gy\zj)} converges uniformly on Uf, and since |rjfc"Hz,)|^|r^"VO (i^->oo) by (6.30), in view of (6.24) and (6.25), we see that {gj''\zj)} converges uniformly on Uj.

§6.1. Theorem of Completeness

297

Put /i, = lim^^ao hi''\ and g;(z^) = lim^^oo gj'^Kzj). Then gj{zj) is a vectorvalued holomorphic function on Uj. By (6.22)

Since |rjfc(z^)|^|r^''^|-»0 (^->oo), taking the limit for ^-»oo, we obtain m

Hence, putting hi""^ = hi""^ - hr, and gY\^j) = gj'^^^j) ~Sj(^j)^ we obtain m

Hence hi''\ r = 1 , . . . , m, and {gj''^} constitute another set of solutions of (6.18) for T — T^''\ But when v tends to infinity, we have hi'^^-^O and ^^PzjeUj \§j''\^j)\-^^, which contradicts L{T)= 1. I We shall prove, using the method of majorant series, that h{s) and gj{Zj, s) converge absolutely for |5|<£ provided that e > 0 is sufficiently small. In general, if two power series of ^ i , . . . , 5/,

i^x,...,i'i = 0

and

are given, we indicate by writing

P(s)«a{s) that |P,,...,J^ «,,...,„

z/i,..., ^; = 0 , 1 , 2 . . . .

As in § 5.3(c), let jb_ -

c'^jSy-h'-'-^sir

A(a)=77^ I —^-^ 1 iC

-2

^

1 6 c j,= \

—.

2

V

By (5.116), we have

Aisf«-A(s). c

b>0,

OO.

298

6. Theorem of Completeness

By induction on v^ we obtain from this inequality the following inequalities: A{sy«\^

A{s),

1^ = 2 , 3 , . . . .

(6.31)

In fact, for j ^ ^ 3, we have A{sY = A(sfA{sr-^«

-A(s)A{sr-^ c

= - A(^)^-^ c

In order to prove the convergence of h(s), gj{zj, s), it suffices to show the estimates h{s)« A(s),

gjizj, s) - zj« Ais),

(6.32)

provided that the constants b and c are properly chosen. For this, it suffices to prove h^is)« Ais),

gjizj, s)-zj«

A(s)

(6.33).

for ?^= 1 , 2 , 3 , . . . . We prove (6.33). by induction on ^ = 1, 2, 3 , . . . . For v= l,sincethelineartermof A(5')is(fe/16)(si + - • -4-5/), the estimate (6.33)i obviously holds provided that b is sufficiently large. Let p^2 and assume that (6.33)._i are established. To prove (6.33). we first estimate Tj]^\^(zj, s). In general, for a power series Pis) = Y.^=i PA^), we denote by [P(s)]. the term P.(5) of homogeneous part of degree P. Then by (6.13), we have TjtUzj, s) = [gj-\fj^{z^,

5), s)]^-[gj^(gr\z,,

5), h'^-'is))]^.

(6.34)

We first^ estimate [gr\fjk(zk, s), s)],. Since the fjk(zk,s) = ^jk(zk) -^Y.7^1 fjk\Azp s) are given vector-valued holomorphic functions, we may assume that fjkizk,s)-bjk(Zk)«Ao(s),

Ao(s)=-^

2 -^—^

^

— (6.35)

holds for ZkE Uj,n Uj with bo>0 and Co>0. Put G(zj, s) = gj'\zj,

s) - Zj.

Then, by hypothesis ofinduction(6.33)._i, we have G(z/, s)« A(^). Namely,

§6.1. Theorem of Completeness

299

letting

A{s)=

I

A,,....,5r> • • • ^p,

we have |G.,...X^,)NA.,...^,

(6.36)

Z,GL/,.

By (6.28), Mo = U j Uj. If z^ G l/f, then since G,^...,^{zj + ^) is a vector-valued holomorphic function of ^ = ( ^ i , . . . , ^„), |^| < 5, it can be expanded into power series in ^ i , . . . , ^„ as

By Cauchy's integral formula, we have

Hence, since |G,^...,X% + ^ ) N A^^...,^ by (6.36), we have

4 8' Hence G,,..,(z,+^)-G.,..,(z,)«A.,..,

I

^;:.:^".

Therefore

For z^-G t/f n (7^, i.e., for Zj = bj^izj,)e Uj, z^e Uj,, we let ^ =fjkizk, Since ^ « Ao(5') by (6.35), we obtain

G(Uz„ s), s) - G(zj, s)« A(s)

I Ml + --- + At„^

Aoisr^^-^^^ =1

8^i^""^^h

s)-Zj,

300

6. Theorem of Completeness

We have

and by (6.31) ^-1

Ao(5). M=l

Remark that (6.35) remains valid if we replace CQ by a larger constant. Therefore, by taking a sufficiently large Co, we may assume that

A
(6.37)

2'

Thus, since Ii^=i (^o/coS)*'~" < 2 , we obtain

%5,(:)(r-«-^il(:)e)'

Ao(5)

-Ao(s), O k

Hence we get Gifjkiz,,, s), s) - G(zj, s)«

-—A(s)Ao(s).

If we take b and c such that b>bo,

c> Co,

(6.38)

we have Ao(s)« {bo/b)A{s). Therefore we get

A(s)Ao(s)«

^0 ^ / N 2 / . ^0 -rA{sr« -A(s). b c

Hence Gifjj^iz,, 5), s) - G(zj, s)«

^—^A{s), Co

§6.1. Theorem of Completeness

301

namely, gr\fjk{zk,

s), s)-fjj,{zk, s)-gj~\zj,

s)-hzj«

r-^A(5). Co

Since by (6.35), fj^izj,, s)-Zj«

Ao(s)« (bo/b)A{s),

gr\fjk{z,, s), s)-gr\zj, s)«

we have

{^-^+J)MS).

Therefore, taking the terms of degree P of the left-hand side of this equality, we obtain the following estimate on Uf n 11^:

[gr'(y;.(z., s), s)]^« {^-^+J)MS).

(6.39)

Next we estimate the second term [gjk(gk~\zk, s), h''~\s))]j, of (6.34). We expand g/fc(^fc + ^5 0 into power series in ^ 1 , . . . , ^„, ^1,. • •, ^m, and let L(^, t) be its linear term. Then, since gjki^k, 0) = bjk(z,^), we may assume that gjk{zk +

^,t)-bjk{Zk)-LU,t) oo-

If we set ^ = gfc~\zfc, 5)-Zfc, and t = h''~\s), A{s) by (6.33)^,-1, we obtain

then, since ^«A(s)

and t«

gAgr\zk,sXh-\s))-bjk{z,)-L{gr\z,,s)-z,,h-\s)) 00

« I

00

aSin-\-mrA{sr«

b\^~^

/

X a^(m + n)'^ -

M=2

M=2

A(s).

\C/

Hence, by taking the terms of degree v of the left-hand side, we obtain [gjkigk

{zj,,s),h

(s)],«

I c

I

^=o\

A(s). c

/

Consequently, taking a constant c such that bao{m-\-n) 1 - ^ -<::, c 2

(6.40)

we obtain [gjkigk

(zj,,s),h

\s))],«

Ais).

302

6. Theorem of Completeness

Combining this with (6.39) we obtain the estimate r,.,|^(z,-,5)«X*A(5),

ZjeUfnU,,

(6.41)

where i^* is given by

c8

b

c

We now estimate rjj^\j,{Zj, s) for arbitrary ZJE Ujn U^. For this purpose, recall that {Tj^iAs)} is a 1-cocycle. Since Mo = [Jj Uj, if ZJE Ujn U^, and Zj ^ Uf, then Zj is contained in some L^f (/ 7^7): Zj = bji(Zi), Zi e t/f. By (6.19) Tjki,(Zj, s) = Bji(Zi)Tik\Azh s) - Bji(Zi)riji^(Zi, s), and, since z^e U^ nUjn (4, both r,fc|^(Zi, s) and ry|^(z„ s) are « K'^Ais) by (6.41). Since we have |B,fc(^k)|<^i with Ki>l, it follows that Tjt,lAzj,s)«2K,K'^A{s), Recall that Ki^s), (6.16),:

r=l,...

ZjE UjnU^.

,m, gj\v{zj, s) are solutions of the equations m

r-=l

Since rjk\j,is)« IKiK'^Ais), by Lemma 6.2, we can choose solutions K\j,(s), r = 1 , . . . , m, {gj\u(s)}, such that Ki^(s)« K.lK.K'^Ais),

gjUs)«

K.lK.K'^Ais).

By definition (6.42) of X*

\

C8

b

C

J

is independent ofp, and if we first choose a sufficiently large b, and then choose c so that c/b be sufficiently large, then we obtain

Note that b and c obviously satisfy (6.38) and (6.40). Hence the above solutions hr\As), {gj\v{s)} satisfy the inequalities K\M«A{s),

g,|.(5)«A(5).

(6.43)

§6.1. Theorem of Completeness

303

Therefore, putting

we see that (6.33)^ follows from (6.43) and (6.33)^-i. This completes the induction, and the inequalities (6.32): his)«A{s),

gjizj,s)-Zj«A{s)

are proved. These inequalities imply that, if \s\ < l//c, h{s) converges absolutely, and gj(Zj, s) converges absolutely and uniformly for ZJE UJ. (c) Proof of Theorem of Completeness In view of ^ = U,- ^ > ^ i = {(^> s) \ \zj\ < 1, j^l < 1} = ^ x D, and Uj = Non WJ=UJ = 0/lfwQput^ = {seC'\\s\

Wj=UjXA.

j

On the other hand, we have M = [J %,

%=UjXBczC"xB.

j

By (6.32) the holomorphic map gj:izj,s)^Uj,t)^{gj(zj,s),h{s)) maps Wj = UjXA into C" xB, and the restriction g,: Uj-^ Uj<^ MQ is nothing but the identity map: (z^, 0)-^(f„0) = (z^, 0). We would like to obtain a holomorphic map g-.X^s^^M by glueing up the holomorphic maps " ' ,gp" ' ,gk," • • But, in general, we may not have g/(Wj)<^%c:C xB. Hence we first restrict g, to gj\%). Since gj(Uj)= Uj, gj\%) is an open subset of Wj containing Uji Uj^gj\%)^Wj,

and gj maps gj\%) into %czJi. For each point ce U^n Uj, there exists a neighbourhood nc) =

{{z,,s)\\z,-c\<8,\s\<8}c:gZ\^,)ng;\%),

304

6. Theorem of Completeness

such that gj and g^ coincide there. Proof. The maps g, and g^ are given by gj: (zj, s) -> Uj, t) = (gj{zj, s), h{s)), gfci (Zfc, s) -> (^fc, t) = (gfc(Zfc, s), h(s)). Note that (z,, s) and (z^, s) are the same point on Jf^ if z^ =fjk{^k, s), while (^j, 0 and (ffc, 0 are the same point of M if Cj = gjkUk, t)- Therefore, to prove that gj and gk coincide on V{c), it suffices to show gjifjkizk, s), s) = gjkigkizk, s), h{s))

(Zk, s) e r(c).

(6.44)

But (6.44) is nothing but (6.10). Hence the power series expansions in Si,..., Si of both sides of (6.44) are the same. On the other hand, both gj(fjk(zk, s), s) and gjkigki^k, s), his)) are the holomorphic functions of Zfc,..., Zfc, Si,..., Si in the polydisk T{c). Hence (6.44) holds. I For each point ce Ujn Uk, choose T{c) as above and let Tjk = Uc "^MThen we have

UjnUkCirjkCzgj\%)ng-k\mk), and the holomorphic maps gj and gk coincide on the domain Vjk. We take a sufficiently small 6 > 0 , and put Uj = {Zje Uj\\zj\
Therefore, putting A^ = {^ G A11^1 < E}, we obtain TT-HAJ-U-f/fxA,,

U^ =

A^<^g;\%),

ufx^,nulx^^czrjk, provided that e > 0 is sufficiently small. Restricting each gj to Uf xA^, we see that gj maps Uf x A^ into Ji and that, if Uf xBn U^x^^y^ 0 , gj and gk coincide there. Hence we can find a holomorphic map

g:7T-\A^) =

UjU!x^s^M

such that g coincides with gj on each Uf x A^. From the expression g: (Zj, s) -> Uj, t) = (gjizj, s), h{s)), it follows that ru ° g = h° TT, while the equality gj{zj, 0) = Zj implies that g is an extension of the identity map go: NQ^ Mo= NQ. This completes the proof of the theorem of completeness: Theorem 6.1.

§6.2. Number of Moduli

305

§6.2. Number of Moduli (a) Number of Moduli In this section we shall prove that the conjecture stated in §5.3 concerning the number of moduli is true in case H^{M, %) = 0. Let M be a compact complex manifold and suppose H^{M, 6 ) = 0. Then, by the theorem of existence (Theorem 5.6), there exists a complex analytic family (Jt, B, m) with OG B e C'" satisfying the following conditions: (i) (ii)

m-\Q) = M. po'. d/dt^{dMt/dtx)t=0'> with Mt = m~\t), is an isomorphism of To(B) onto H\M, 0 ) : To(B) ::>^o H\M, 6 ) .

By the theorem of completeness (Theorem 6.1), (M, B, m) is complete at OeB. We need to prove that m{M) = dim H\M, %) holds provided m{M) is defined. Suppose that m{M) is defined and put /x = m{M). Then, by Definition 5.4 of the number of moduli (p. 228), there exists a complete complex analytic family (^, D, TT) which is effectively parametrized and such that 7r~^(0) = M, with D being a domain of C^ containing 0. We denote by 5 = ( ^ 1 , . . . , 5^) a point of D. Since {Jf, D, IT) is complete, it follows that, if we take a sufficiently small domain A, OG A c B, then {M^,^, tjr) is a complex analytic family induced from {Jf, D, TT) by a holomorphic map h: t-^s = h(t) with h(0) = 0. Hence M^^ Nh(^t^ = 7T~\h{t)). Therefore, by (4.30) — r = I —•^—,

( 5 i , . . . , 5 ^ ) = 5 = /z(r).

Since po- To(B) ^ H\M, S) is an isomorphism, the cohomology classes

(— )

eH\M,e),

A = l,...,m,

m=

dimH\M,e)

on the left-hand side of this equality are linearly independent. On the other hand, we have /JL = m ( M ) ^ dim H\M, 6 ) . Hence we obtain the equality m(M) = dim H\M, O ) . Thus we have obtained Theorem 6.2. Suppose H^(M, 0 ) = 0 and that the number of moduli m{M) ofM is defined. Then m(M) = dim H\M, 0 ) . Then what is the condition for m{M) to be defined in case H^(M, 0 ) = 0? If m{M) is defined, (Jt^, A, tir) is, as stated above, the complex analytic family induced from an effectively parametrized, complete family (Jf, D, TT) by a holomorphic map h: A-»D with /i(0) = 0, and m(M) = dim H\M, 0 ) .

306

6. Theorem of Completeness

Since D is an effective parameter space, the linear map d/ds-^dNJds is an injection of T^iD) into H\NS,S). Since dim T^iD) = m(M) = m, we conclude dim H^(M,&)^m. On the other hand, dim H^{No, SQ) = dim H^(Ns,Ss) — m. Therefore, by the theorem of upper-semicontinuity (Theorem 4.4), dim//^(N^, 0 J = m for any s satisfying \s\<8 provided that s is sufficiently small. If we take a sufficiently small domain A with 0 G A c= B, we have M, = Nh(t), and \h{t)\ < e for any t e A. Hence we conclude dim H\Mt, ©,) = m. Thus, if m{M) is defined, then dim H\Mt, 0,) = m is a constant independent of t. Conversely, if dim H^(M^, 0^) = m does not depend on re A, for a sufficiently small domain A, 0 e A c: J5, then m{M) is defined. Proof. For t = 0, l^-^)

\ dt^ / t=o

G//HMo,eo),

A = l , . . . ,m

are linearly independent. Therefore, if m(M) = dim H\Mt, dent of t, it follows that

^-^eH\M,S,\

0^) is indepen-

A = l,...,m

are linearly independent for \t\<8 provided that e is sufficiently small. This fact will be proved in the next chapter (Theorem 7.11). If we simply write A instead of A^ ={rG A||f|< e}, then (M^,^,m) is an effectively parametrized complex analytic family, and p^: r^(A)->/f ^(M^, 0^) is surjective for f G A. Consequently, by the theorem of completeness (Theorem 6.1), the complex analytic family ( ^ A , A, ixr) is complete. Hence the number of moduH of M = XJJ~^{0) is defined and equals m. I Thus we have proved the following Theorem 6.3. Suppose JF/^(M, 0 ) = O. Then the number of moduli m{M) is defined if and only if dim H ^ (M^, St) is independent ofteA when A, 0 G A c= B, is taken sufficiently small. If this condition is satisfied, then (Ji^, A, m) is an effectively parametrized, complete family. In case H^(M, 0 ) = 0 holds, which turns out to be true in many examples, the following theorem holds. Theorem 6.4. IfH\M, 0 ) = H\M, 0 ) = 0, then m(M) is defined and equals dim H\M, 0 ) : miM) = dim H\M, 0). Proof. Consider a complex analytic family (M, B, ur) satisfying the above conditions (i) w~'{0) = M, and (ii) To(B) S H\M,e).

Since /f^(M,0) =

§6.2. Number of Moduli

307

H\M,&) = 0, we have, by Theorem 4.4, dim//^(M„ 0,) = dimH^(Mt,St) = 0 for r e A provided that A with OeAczB is sufficiently small. In general, if dim H^~\Mt, 0^) and dim H'^^^Mt, 0^) are independent of t, so is dim H'^iMt, St). The proof of this fact will be given in Chapter 7 (Corollary to Theorem 7.13). Therefore, in the present case, dim H^{Mf, St) is also independent of t. Hence, by Theorem 6.3, m(M) is defined and, by Theorem 6.2, m(M) = dim H\M, S ) . I Moreover, by Theorem 6.3, (Jt^, A, m) is effectively parametrized and complete. Thus, if H^{M,(d) = H^(M,(d) = 0, M has an effectively parametrized complete family (Jt^^, A, m) with 7r~\0) — M. By studying several examples of compact complex manifolds, we have observed a phenomenon that m{M) coincides with dim H\M,S) for these examples. It is in order to explain this fact that we have so far developed the theory of deformations, and we finally obtained Theorem 6.4, which makes clear that the equality m{M) = dim H\M, 0 ) holds if H^{M, 0 ) = H^{M, 0 ) = 0. This might be considered a milestone of our theory. But the fact is that there are many examples for which //°(M, 0 ) = O holds, but H^(M, 0 ) 7^ 0. So we have to say that Theorem 6.4 can only apply to the very limited class of compact complex manifolds. (b) Examples (i) Quadratic transforms of P^. We consider the quadratic transforms of M =

Q,^"'Q,^Q,SP')

as in §5.2(a)(iv). As we have seen in §5.2(a), the number m of apparently effective parameters used in constructing M is equal to dim H\M,(d). Since H\M,(d) is 0 for v^A, we suppose v^5. Then JF/^(M, 0 ) = H\M,e) = Q (see p. 225). Therefore, by Theorem 6.4, m(M) is defined and m(M) = dim f/^(M,0). Thus the number of parameters m coincides with the number of moduli m{M). (ii) Quartic surfaces. Let M^ be a non-singular hypersurface of degree h in P""^^ where we suppose n ^ 2 and h^l (see p. 219). We have seen in §5.2(c) that, except for the case n = 2 and /i = 4 , m(Mt) is defined and the equality m(Mt) = dim H^(Mt, St) holds (see p. 247). For the case n = 2 and /i = 4, that is, for the quartic surfaces Mt in P^, it remains unsettled whether the number of moduli m{Mt) is defined or not. By using Theorem 6.4, we can show as follows that the number of moduli m{Mt) is also defined and equals dim H\M, 0 ) in the case of quartic surfaces. By (3.157) dim H\Mt,

0,) = dim H%Mt,

n\Kt)),

308

6. Theorem of Completeness

where Kf denotes the canonical bundle of M^. Let E be the line bundle over P^ defined by the transition functions C^^k)/Cinj) (see p. 175), and let Et be its restriction to M^. Then, for a non-singular hypersurface M^ of degree h in P^, we have K, = E':-*

(6.45)

(see p. 178). In the present case, since /i = 4 , Kt is trivial. Hence we have dim ff^(M„ e,) = dim

H^'iM^.a]),

while H ^ ( M „ a J ) is 0 (see p. 179). Hence /f^(M„0,) = O. On the other hand, H^(M„ @,) = 0 (p. 245). Hence, by Theorem 6.4, m{Mt) is defined, and since dim H\Mt, St) = 20 (see p. 247), we have m{Mt) = dim H ^ M , 0,) = 20. This justifies what we stated in p. 247. The equality dim H\Mt, @ J = 20 also follows easily from the RiemannRoch-Hirzebruch theorem (5.24). Since Kt is trivial c, = -c{Kt) = 0,

n'=€(^/\

T * ( M ) ) = O{Kt) = 0,

Hence p^ = dim//^(M^O^) = d i m / / ^ M , , ^) = 1. Since dim//^(M„ 0 , ) dim H\Mt, 6,) = 0, we have by (5.24) that dim H\Mt,&t)

=lc2-

As q = dim H^(Mt, fl^) = 0, we have Pa=Pg-Q=^' Consequently, by Noether's formula, we have C2 = 24, and hence dim H\Mt, 0^) = 20. (iii) Surfaces of arbitrary degree. We shall compute dim H^{Mt, 0 J for a non-singular surface of degree / z ^ 3 , /i7^4, in P^. Since, by (6.45), Kt = E^~'^, €(Kt) is the restriction of OiE^'"^) to M,. If we denote the restriction map by r,, then, since [Mt] = E^, we have ker r^ = 0(E''~^®[MtT') = €(£-''). Hence 0^€(E-'')^O{E''-'')^^O(Kt)->0 is exact. We note that, in the corresponding exact sequence of the cohomology groups 0-^H%P\ €(E-^)) -> H\P\ -^H\P\€{E-''))-^"',

^ ( E ' - ^ ) ) ^ H%Mt, €{Kt))

§6.2. Number of Moduli

309

we have, by Bott's theorem H\P\ €{£-'')) = 0. Hence H\P\

(Theorem

OiE""-^)) = H\M„

5.2),

H\P\

OiK,)).

OiE'"^)) =

(6.46)

As is easily seen from the proof of Lemma 5.4, for any natural number k, dim H^(P^, 0(E^)) is equal to the number of monomials of degree k in ^o, fi, ^2, ^3, that is Ct^). Hence, by (6.46)

p,=Hh-l)ih-2)(h-3). Next, since dim H^(Mt, ©,) = 0, using (5.24) and (5.25), we get dim H\M,,

e,) - dim H\M„ S,) = 2c\ - 10(/7, + 1).

by (6.45), while p^^Pa Since c{Etf=h (p. 180), we have c\ = {h-Afh because q = dim H^{Mt, (1^) = 0. Consequently we obtain dim//^(M„e,)-dim//HM,@,) = 2(/i-4)2/i-f(/i'-6/z^+ll/i). (6.47) Since we have by (5.21) dim H\M„

6,) = l{h + 3){h + 2)(h + 1) - 16,

h9^4,

(6.48)

we conclude dim//^(M„e,)=i(/i-2)(/z-3)(/i-5),

/i7^4.

(6.49)

In particular, H^(Mt, ©^) T^ 0 for /i ^ 6. Thus Theorem 6.4 does not apply to these cases and cannot explain the fact that m{Mt) coincides with dim H\Mt, 0^) in these cases. As stated before, for a hypersurface M of an Abelian variety A""^^, the number of moduli m(M) is defined and the equality m(M) = dim H\M, 0 ) holds (p. 248). Using the Riemann-Roch-Hirzebruch theorem, we can compute dim H^(M, ©) for a surface M in A^. The result is dim H^{M, S) = 3e + 5, where e is a suitable natural number (see [21]). Hence we always have H^{M, 0 ) T^ 0 in this case. (iv) Consider the complex analytic family constructed in (2.33) in case m = 2 and h = l. Employing the same notation as in Example 2.16, we have M , = U^xP'uU2XP\

Ui=U2 = C,

where (zi, (i)e U^ xP^ and (z2, ^2)^ i^2XP^ are the same point of M, if

310

6. Theorem of Completeness

and only if z,Z2=U

^i = zU2-^tZ2.

(6.50)

We have seen that MQ = M2, and M, = Mo = P^ xP^ for r 7^ 0 (p. 73). Recall that

M^=U,xP'uU2XP\ where {z^, ^i)e U^xP^ and (22, ^2)^ ^iXP^ are identified if ZiZ2= 1 and fi = ^r^2. To compute dim H\M^,e), we first prove H\CxP\e) = 0. Proof. Since P Ms a union of Ui and U2, C x P ^ is written as CxP'=yiuy2,

where

V, = CxU,,

V2 = CxU2,

i.e., '^ = {Vi, V2} forms an open covering of CxP^ by two coordinate neighbourhoods Vi and V2. Both Vi and V2 are biholomorphic to C^ = C x C, and Dolbeault's lemma (Theorem 3.3) is valid for C^ regarded as a bidisk of radius 00. Hence Lemma 5.2 applies to the covering 93, and we obtain

H\C XP\ O) = H\^, S) = ZH«, e)/8C%^, 0). A 1-cocycle {^12, ^21} belonging to Z^(SS, S) consists of holomorphic vector fields 612 and ^21 on V^ n V2 with ^12 = -^21, while C^(9S, 0 ) is the set of all pairs {^1, 62}, where Oi and 62 are holomorphic vector fields respectively on Vi and V2. Hence, in order to prove //^(9S, 6 ) = 0, it suffices to show that any 1-cocycle {612, ^21} is represented as 8{62, O2}, i.e.,

Let w be the coordinate on C. Then, in terms of the coordinates (w, Zi) on y^ = u^ x p \ the holomorphic vector field O12 on V^ n V2 is written in the form Oi2=u(w, z i ) — + u(w, Zi) — , dW

dZi

where M(W, Zi) and i;(w, Zj) are holomorphic functions on VinV2 = C X (L^i n L^2) = C xC*. Hence they can be expanded into Laurent series in Z\' +00

W(W, Z i ) =

I n = —00

+00

Wn(w)z",

U(W, Z i ) =

I

l^n(>v)z",

n = —00

where the u„(w), v„iw) are entire functions of w. In view of Zi = I/Z2 and

§6.2.

Number of Moduli

311

d/dzx = -zl{d/dZ2), if we set

n=0

OW

n =0

OZi

+00

•)

+00

n

„=1

dW

„=i

dZ2

62=1 «_„(w)zs—-1 D_„(w)zrv> then ^1 and O2 are holomorphic vector fields on Vi and V29 respectively, and satisfy 612 = O2-O1. I FutWj=UiXP^ and W2 = ^2 xfP^ Then 28 = {\^i, W2} is an open covering of Mrn, and both W^ and W2 are biholomorphic to C x P ^ Hence H\ Wu 0 ) = H'( W2, 0 ) = 0. Therefore, by Theorem 3.4, we have

H\M^,

e) = H\m, e) = z^ss, e)/8c%m, e).

We represent a 1-cocycle {612, ^21} ^^^(SB, 0 ) by the holomorphic vector field (9i2 = -^21 on W^ n W2. Then, this 1-cocycle belongs to 5C^SB, 0 ) if and only if there exist holomorphic vector fields ^1 and 62 respectively on Wi and W2 such that ^2-01 = ^12.

(6.51)

By (2.36) di is written in the form ^i = t^i(^i)/-+(«i(^i)^?+iSi(z0^i + r i ( z O ) - f , dZi

dL,x

where i^i(zi), ai(zi), ^i(zi), yi(zi) are entire functions of Zi. Similarly, by (2.37), 62 is written in the form ^2=t;2(Z2)-^+(a2(Z2)^^+i82(^2)^2+r2(^2)):;7-, ^^2 dCi where 1^2(^2)? (^li^i)^ Pii^i), 72(^2) are entire functions of Z2. We write 612 as follows in terms of the coordinates (zi, ^i): ^i2 = t;i2(zi)-^+(ai2(zi)f? + i8i2(^i)^i + ri2(^i))T7r. Here 1^12(2:1),* «i2(^i)? iSi2(^i), 712(^1) are holomorphic functions of Zi on U^n 1/2 = C^, hence, are expanded into Laurent series in z,. In terms of the coordinates (zj, fj) with Zi = l/z2, ^i = z^^2 in place of (z2, ^2), ^2 is

312

6. Theorem of Completeness

written in the form

dZi

\

Zi

/ dii

(see p. 74). Hence the equation (6.51) is reduced to the following system of equations: -z^^V2\—j - v^{z^) = Vi2{zi),

zToi2(—]-^\izi)

= ai2{zi),

m z i i ; 2 f - ) + i 8 2 ( - j - i 8 i ( z i ) = iSi2(2i), -i^^ril —) - yi(zi) = yiiizi). Z\

\Zx}

In ca^e m = 0 or 1, these equations always have a solution. For m^2, let 712(^1) =Z^=-oo ^n^" t)e the Laurent expansion of 712(^1)- Then the above equations have a solution if and only if c_i = c_2 = • • • = c_^+i = 0. Hence we obtain dim

H\M^,e)

m = 0, 1, m = 2,3,4,...,

(6.52)

and for m ^ 2 , the 1-cocycles e,2 = z'2 — eZ\^,e),

fc=l,2,...,m-l,

(6.53)

form a basis of H\M^, @) ^ H^SB, 6 ) . Next we shall prove (6.54) Proo/ Since we have dim H\M^,@) = dim H\Mrn,^\K)) by (3.157), it suffices to prove /f^(M^, a ^ i ^ ) ) = 0. Since M^= U^xP'^ U2XP\ P^ = C u {00}, any (/r e H^{Mm, ^\K)) is written on L/i xC c= [/i xP^ in the form ^ = {g(zu Cx) dz, + /z(z„ ^1) dC,)®{dz, A c/^0,

§6.2. Number of Moduli

313

where g(zi, fi) and h{zi, ^i) are holomorphic functions of Zi and ^j 7^00. We note that if/ is required to be holomorphic in a neighbourhood of Ui x 00. Changing the coordinate ^1 to the local coordinate w = l/^i at 00, we obtain

Thus (l/>v^)g(zi, 1/w) and (l/w'*)/i(zi, 1/w) must be holomorphic in w. Hence g(zi, ^0 = h{zu Ci) = 0, i.e., ip = 0, I With these preparations, we return to the study of the complex analytic family defined by (6.50). Since MQ= M2, and M^ = MQ, t9^0, WQ have, by (6.52),

dim/fHM„€),) = |^'

J^^'

(6.55)

In order to compute the infinitesimal deformation of M^, we write (6.50) as Zl =f\2{Z2, ^2, 0,

Cl =fUz2, ^2, 0,

where f^izi, Ci, 0 = 1/^2 is independent of t^ and 7^2(^2, ^2, 0 = ^2^2+ ^^2The infinitesimal deformation dMJdteH\Mt,%t) is nothing but the cohomology class of the 1-cocycle . .,. a/l2(^2, ^2, t) 0 Jt) =

d

= Z2

d

.

For 1-0, Z2{d/d^i) forms a basis of / / ^ M o , BQ) = / / ' ( M 2 , 0 ) by (6.53). Hence po- To(C)^ H\MO,SO) is surjective. Therefore, by the theorem of completeness (Theorem 6.1), the family {Mt\teC} is complete at ^ = 0. Since H^(M2, B) = 0 by (6.54), there exists a family (M, B, m) such that zjj~\0) = M2 and that poi To{B) ^ H\M2, 0 ) with OeBaC. This is nothing but our {MjrGC}. For r^^O, H\M,, 0,) = O by (6.55). Hence dMJdt = 0 and the family {MjreC} is complete at any t. Therefore { M J ^ G C } is a complete complex analytic family. We can also prove dMJ dt = 0 for r^^O by direct calculation. By (6.50) = dZ2

5 zldz,

+(2Z2^2+0 , 'dC,

=zl 8^2

Hence d tZ2—=Z2d(i

d dZ2

^^ d d 2^2 — + ^ ! — . 3^2 dZi

dCi

314

6. Theorem of Completeness

Therefore, if we put

t \

dZ2

3^2/

6i{t) and ^2(0 are holomorphic vector fields on (7i xP^ and L/2 xP^ respectively, and we have ^12(0 = ^ 2 ( 0 - ^ 1 ( 0 . Hence, for t^O, the cohomology class dMJdt of the 1-cocycle ^12(0 is 0. Note that the limit ^i(O) = lim.^o ^i(0, ^2(0) = HnifUo ^2(0 do not exist. This explains the reason why dMJdt 7^ 0 for t = 0, even though ^12(0 = ^li^/^Cx) is apparently independent of t. For the complete family {M^ U e C}, we have dim H\MO, SQ) = 1, and dim H\Mt, @t) = 0 for r 7^ 0. This implies, by Theorem 6.3, that the number of moduli m(M2) is not defined. Recall that dim H^{M2, %) = ! (see p. 75). This example shows that even if H^{M, 0 ) = 0, the number of moduH m(M) is not necessarily defined in case H^{M, 6 ) 7^ 0. The complete family {Mt\teC} contains all sufficiently small deformations of M2 = ^0 (P- 228). Therefore for the study of small deformations of M2, it suffices to investigate {M, | r e C}. For r 7^ 0, M, = MQ = P^ x P \ so Mt "jumps" from M2 to MQ at the moment t moves away from 0, and M^ does not change as far as ^ 7^ 0. This suggests that the number of parameters for M2 is not 0, but "almost" 0. The existence of such examples as M2 above is one of the reasons why we do not define m(M) if there exists no effectively parametrized complete family {M, B,m) with tt7~^(0) = M. Let {Jt, B,xu) be a complex analytic (or differentiable) family and let Mt = m~\t). Suppose that H^MfO^ 0^0) = 0 at some point t^eB. Then, by the Frolicher-Nijenhuis theorem (Theorem 4.5), Mt = M^o for all t sufficiently near t^. In other words, the complex structure of M^o does not vary by a small change of the parameter t. But for a large deviation of t from t^, Mt is not necessarily biholomorphic to Mt. The above-mentioned family { M J ^ G C } gives such an example. Namely, if we take ^ ^ = 1 , then

M, = P'xP\

H\M,,e,)

= 0,but

Mo=M2^M,.

§6.3. Later Developments We have explained above how the theory of deformations of compact complex manifolds developed till about 1960. We observed the phenomenon that for several compact complex manifolds M, their number of moduli m{M) coincides with dim H\M,S), and conjectured that the equality m(M) = dim H^(M, B) might hold for any compact complex manifolds. Considering this as a working hypothesis, we developed the theory of

§6.3. Later Developments

315

deformations and proved that the conjecture is true if H^{M, 0 ) = //^(M, (2)) = 0 (Theorem 6.4). However, with many examples, we find that H^(M, S) 9^ 0. For example, as in (b) of the previous section, H^(M, 0 ) ?^ 0 for a non-singular surface M of degree at least 6 in P^. In spite of this, the equality m(M) = dim if ^(M, O) holds for these surfaces. At that time we could find no example of M for which m{M) 9^ dim H\M, 6 ) . In order to explain this strange phenomenon, we thought that we need to generalize the theorem of existence (Theorem 5.6) to the case H^{M, 0 ) 9^ 0. As stated earlier (p. 259), there exists an M with 0e H\M, 6 ) such that [0, e]9^0. Therefore, in general, there does not exist, for a given M, a complex analytic family (Jt, B, m) satisfying the conditions (i) and (ii) of Theorem 5.6. But at that time—about 1958—we could not understand at all what should be the conditions for the complex analytic family that we must seek. We did not know what is to be proved to exist. H. Grauert, who was working very energetically on several complex variables, agreed with us that this was a difficult problem. The theorem of existence for general case was proved by M. Kuranishi in 1961 ([22]). We will explain the outline of his theory below. We first recall the equality (5.105):

if^'^(T) = H'''^(r)en^^'^(r),

T=T{M).

Substituting H^'^ (T) 0 D if^'^ (T) for if^'^(r) in the right-hand side, we obtain, in view of H^'^(r) = 0,

if''^(r) = H''^(r)enn^''^(T). Hence any ijf in ^^'^(T)

(6.56)

is written in the form

Since H^'^(T) and nif^'^(T) are orthogonal to each other, 77 and (p are determined uniquely by if/. Writing Hif/ for rj and Gif/ for (p, we obtain ^ = Hijj + nGip,

HiljeH'''''(T),

GilfeD^'^'^T).

(6.57)

Note that H: if/^ Hif/ and G: il/-^ Gip are linear operators. We call Hijj the harmonic part of J/^, and G the Green operator. The Green operator that was introduced in §5.3(c) is a special case of this one. Since we have H^ijj = 0, it follows by (6.57) that ^(A = D Gdijj. On the other hand, applying 'd to both sides of (6.57), we obtain dijj = dUGilf = dbdGil/ = UdGiff

316

6. Theorem of Completeness

because D =ab + ba. Hence we obtain Gd = dG.

(6.58)

For this G, we have the following estimate similar to (5.111): |G^U^«^c^|(AU-2+«,

ilfe^'-'iT),

k^l,

(6.59)

where c^ is a constant independent of ifj. Let {^u . . . , /3^} be a basis of H^'^T) and put

We shall determine a power series

j^lH

Hi/^^2

in ? i , . . . , ^m, hy requiring the condition
(6.60)

Writing (/?(0 =
hG[
^= 2,3,....

i

Hence ^ ( 0 is uniquely determined by the condition (6.60). By the same method as in §5.3(c), we can prove that (p{t) converges with respect to the Holder norm | \k+a for t with |^|<e, provided e > 0 is sufficiently small, and that (p{t) is C°° on M x A „ where ^,={teC'^\\t\<e}. Lemma 6.3. Suppose e>0 is sufficiently small and |^| < e. Then (p(t) in (6.60) satisfies the integrability condition (5.86): ~d
§6.3. Later Developments

317

Conversely, suppose that H[(p,(p] = 0, and put il^=^(p, (p] — d(p. Since d(pi{t) = 0, applying d to both sides of (6.60), we obtain d(p=^t>G[(p,(pl Since H[(p, (p] = 0, [)G[(p, (p] = \t>dG[(p, (p] = ^hGd[(p, (PI Using (5.88) and (5.87), we obtain d[(p, cp] = 2[d(p, cp] = [[(p, (p\ (p'] - 2[(/^, (p\ while [[(p, = -bG[iA, (;p]. By (5.118), (5.119), and (6.59)

Hence

If e is sufficiently small and if |^|<e, then KiK2C^\(p{t)\k+cx<\' Hence I'Alfc+a <\\^\k+cx, and (/^ = 0. Thus we obtain ^cp =^(p, (p\ I Let { r i , . . . , 7/}, / = dim //^(M, 6 ) , be an orthonormal basis of H^'^( T). Then we can write H[cp{t),ip{t)^=

i

{[cp{t),^{t)ly^)y^.

k=\

Put fk{t) = ([(P(OJ <^(0]J 7k)- Then the /^(O are holomorphic functions of t. By»Lemma 6.3, <^(r) satisfies the integrability condition if and only if fi{t) = ' ' ' =fiit) = 0. Therefore, if we set B = {teA^\Mt)

= "'=fM

= 0},

then B is an analytic subset of A (p. 33), and, by the Newlander-Nirenberg theorem (Theorem 5.5), (p{t), for each teA, defines a complex structure Mt on the differentiable manifold M. In this way, we obtain a ''complex

318

6. Theorem of Completeness

analytic family" {Mt\teB} of deformations of M. But, note that in this case B may have singularities so that we must suitably generalize the definition of "complex analytic family". Furthermore, the family {M^ \te B} can be proved to be complete in a sense similar to Definition 5.3 ([22]), but the proof is rather difficult. Theorem 6.5 (Theorem of Existence). For any compact complex manifold M, there exists a complete complex analytic family {Mt\te B}, 0 G J5 c: A^ with Mo = M. The above family {Mt\teB} is called the Kuranishi family. Here B is an analytic subset of A^ defined by / holomorphic equations / i ( 0 = * * ' = fi(t) = 0, where dim ^, = m = dim H\M, 6 ) , and / = dim H\M, O ) . Hence, we have dim B ^ dim H\M,

S) - dim H\M,

0).

(6.61)

B ) - dim H\M,

6),

(6.62)

From this inequality, it follows that m{M) ^ dim H\M,

provided m(M) is defined. If H^{M, 0 ) = 0, Theorem 6.5 is reduced to Theorem 5.6, herice Theorem 6.5 is an extension of Theorem 5.6. But, if we take the view-point of trying to explain the phenomenon that the equality m{M) = dim H\M,S) holds for various examples of M, then Theorem 6.5 is quite different in nature from Theorem 5.6. For example, let M be a non-singular surface of degree h in P\ Then, by (6.47) dim H\M,

0 ) - d i m H\M,

0 ) = —(/z^-18^1 + 41).

Hence the left-hand side is negative when / i ^ 16, and in these cases the inequalities (6.61) and (6.62) are trivial. Thus in case dimf/^(M, 0 ) ^ dim /f ^(M, 0 ) , Theorem 6.5 is an existence theorem for its own sake, and gives us no information about the family {Mt\te B} except that it is complete. In particular. Theorem 6.5 is ineffective in explaining that m{M) coincides with dim H\M, 0 ) for various M. Nevertheless, it is obvious that Theorem 6.5 is of fundamental importance in the theory of deformations of complex structures. This reflects a fundamental difference between mathematics and physics. In physics, if a theory is developed for the purpose of explaining certain phenomenon and fails to do so, then that theory must be useless. Even with Theorem 6.5 in hand, we could not prove our conjecture m(M) = dim/f^(M, 0 ) , so we suspected that there might exist counter-

§6.3. Later Developments

319

examples to our conjecture. In search of such counterexamples, we calculated m(M) and dim H\M, 6 ) for various algebraic surfaces. We believed that algebraic surfaces were relatively easy to handle, but calculating m{M) and dim H\M, S) was difficult in most cases. Furthermore, to our surprise, d i m / / ^ ( M , 0 ) was much more difficult to calculate than m{M). The is determined only by M itself, while, in order to dimension of H\M,S) calculate m(M), we must know an effectively parametrized complete complex analytic family (M,B,m) with m~\0) = M. So we supposed that dim H^(M, 6 ) would be easier to calculate than m{M), but it turned out to be the contrary. For many examples of M, while m{M) was calculated, we could not find dim H\M,S). Still, when we got both of m(M) and dim H\M, ©), we found that miM) = dim H\M, 0 ) holds. In 1962, D. Mumford constructed a counterexample to this conjecture ([25]). His example is a 3-dimensional complex manifold M = iJidP^) obtained from P^ by a monoidal transformation fjuc whose centre is a certain curve C c: p^ of genus 24 and degree 14. Because Mumford's construction cannot be applied to the case of surfaces, we hoped that the conjecture might be true for surfaces, i.e. for 2-dimensional compact complex manifolds. However, in 1967, A. Kas found a 2-dimensional counterexample. A surface M^ is called an elliptic surface if the following two conditions are satisfied: (i) There exists a holomorphic map : M ^ ^ A of M onto an algebraic curve, i.e., a compact Riemann surface A. (ii) For a point w G A, the inverse image C„ = <^~\u) is an elliptic curve except for a finite number of points M = ^ i , . . . , a^. (For details on elliptic surfaces, see [15].) For a general elliptic surface M^, the equality m{M) = dim H\M, 0 ) holds, while, for any pair of natural numbers k and g satisfying g>24/c —2, Kas constructed an elliptic surface M^ with m(M^) = 3A: + 4 g - 3 , d i m f / H M ^ 0 ) = llA: + 4 g - 3 ([13]). As to this elliptic surface M^, A = (M^) is a curve of genus g, and there exist 2k points a^, A = 1 , 2 , . . . , 2A:, on A such that, for uiA a^, C = C^ is a fixed elliptic curve not depending on u. Thus M^ is a certain special elliptic surface with rather simple structure. In this way, the conjecture m{M) = dim H\M, 0 ) ultimately turned out to be false. However, because of the facts that this conjecture played an important role as a working hypothesis in the development of the theory of deformations and that it was quite difficult to find counterexamples, and so on, we supposed that those M for which m ( M ) < dim H\M, 0 ) holds might have some special structure, and that for general M, m(M) would coincide with dim H\M, 0 ) even if H^{M, 0 ) 9^ 0. But we could not succeed in finding a simple and useful criterion for the equality m{M) =

dim

H\M,e).

Chapter 7

Theorem of Stability

In this chapter we give proofs of Theorem 4.1, Theorem 4.4, Lemma 4.1, and other results used in the previous chapters. The proofs depend on the theory of differentiable families of strongly elliptic differential operators.

§7.1. Differentiable Family of Strongly Elliptic Differential Operators (a) Strongly Elliptic Partial Differential Operators Let X be a compact differentiable manifold and { Uj} a sufficiently fine open covering of X where each Uj is a coordinate neighbourhood. Let Xj. x-^ Xj = ( x j , . . . , xJ) be a system of C°° local coordinates of X defined on some domain containing [Uj]. We assume that X is oriented and that its orientation is determined by the coordinate system {Xj}. Let B be a complex vector bundle over X, TT its projection, and ( f ] , . . . , ^/) the fibre coordinates of B defined over some domain containing [Uj]. If we let bj^ix) = {bjk^(x)} be a system of transition functions of B, then 7r~\Uj)= UjXC, and the points (x, Cl..., ^;) G Uj x C " and (x, ^L . . . , Ck) eUkXC" are the same point on B if and only if ij=Y,^ ^^v^fcA C°° section il/ of B over X is represented on each Uj by a vector-valued C^ function iff = IIJJ{X) = {if/jix),..., IIJJ(X)), where lAt = Z ^ b^k^^'j^ holds on Ujn Uk9^ 0 . Let L(B) be the vector space of all C°° sections ip of B over X, and E a linear partial differential operator of L{B) into itself. Namely, E: if/-^ Eij/ is a linear map of L(B) into itself such that we can write, on each Uj, (Eik)^{x) = i

Elix,

Dj)^f{x),

A = 1,..., K

Here Ej^(x,Dj) is a polynomial of Dj^=d/dxf,a coefficients are C°° functions of x:

= l,...,n,

(7.1) whose

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

321

We express Ej^{x, Dj) as a sum of homogeneous polynomials Ef^{x, Dj) of degree / in Dj^, a = 1 , . . . , n. Then (7.1) is written as m

u

iE,p)fix) = 1 1

£j^(x, Dj)il,f{x),

A = 1,..., K

(7.2)

/ = 0 fx = l

Here we may assume that at least one of EfJ^{x, Dj) is not identically zero. If so, m is called the order of E. In what follows, we assume that the order of E is even. Let gacfi be a C°° symmetric covariant tensor field of rank 2 on X (p. 106). On each Uj, g«^ is represented as Z

gjap(x)dxf

dxf,

where

dx^ dxf

=\{dxf®dxf-hdxf®dxp,

and the gjapM = gjpaM are real C°° functions on Uj. If the corresponding quadratic form Z«,^ g7«/3(^)^"f^ in real variables ^^ . . . , f" is positive definite at each point xeX, then X« ^ gjap M dxf dxf is called a Riemannian metric on X In this case, if we set gj = gj(x) = det(gj«^(x))«^^^i... „, we have gj(x)>0, and V g / ( x ) J x j A • • • A dXj =\/gkix)

dxi A • • • A JXfc

on Ujn Uj^T^ 0 . For simplicity, we use the following notation: Vg dX = Jgj dXj = ^/gj(x) dx]A"'A

dxj.

Then Vg J X is a C°° n-form on X, and vgy(x) > 0. Hence v g J X is a volume element of X which is invariant under coordinate change. Now we shall introduce an inner product on L(B). For this, we assume that X has a Riemannian metric Y.l,p=i gjafiM dxj dxj_, and that, on the vector bundle B, a Hermitian metric Zl,^=.i ^jAfiMCj^j" is defined on the fibres. For any lA, ^ e L(B), we define their inner product by r

X aj,^ix)4>Hx)
(7.3)

We denote by \\^|/\\=^/(^|J, ifj) the norm of IIJ. Definition 7.1. Let £ be a linear partial differential operator. If, for any (p,il/eL(B), the equality {Eip, (p) = (il/, Ecp) holds, then E is said to be formally self-adjoint.

322

7. Theorem of Stability

If E is formally self-adjoint, then as is easily verified by integrating by part, we have the following equality. Z aj^^{x)E^,-ix,Dj)=

I

a,,^{x)El%x,Dj).

(7.4)

In the sequel, E is not necessarily assumed to be self-adjoint, but we always assume that the equality (7.4) holds for E. Consider the polynomials E'P^ix, ^j) in ^ ^ i , . . . , ^j„ which are obtained from £j^"(x, Dj) by replacing Dj^ =d/dxj by the real variables ^„ = ^^^. We regard ^« as the components of a covariant vector ^ E T J ( X ) at X G X Put

AU4c<)=(-i)""' I v^W£,r(x, f,)^;^;. Then, by (7.4), Aj{x, ^j, {,) is a Hermitianform in ^ ] , . . . , ^/. Since E^A ^ ^ ( ^ ) for any if/e L(B), we have, on L/^ n Uk^0, (Eilj)}(x)=lbj,^(x)(Eilj)Ux), hence

Therefore, we have, on UjO 17^7^ 0 ,

Hence, suppressing the subscript j , we may simply write A{x,^,n

= Aj(x,^j,Q.

(7.5)

Let (g"^(x))«^^ =.!...„ be the inverse of the matrix (gai3{x))a,i3=i,...,n. We define the length of a covariant vector ^e r J ( X ) by |f| = V Z l ^ ^ g ^ ^ W ^ . Definition 7.2. A linear partial differential operator E of even order m is said to be strongly elliptic if there exists a constant 5 > 0 such that, for any xeX, ^e r * ( X ) , and f e B^, the following inequality A(x,^,a^d'\^r

Z

aj.^ix)^}^,^

(7.6)

holds. In the rest of this section, we assume that E is a strongly elliptic, formally self-adjoint linear partial differential operator.

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

323

Let F be the linear subspace of L(B) consisting of all solutions of Eif/ = 0. It is known that ¥ = {il/e L{B)\ Eij/ = 0} is of finite dimension (for the proof, see Theorem 7.3 in the Appendix). Moreover, L{B) is decomposed as follows into the direct sum of the subspaces orthogonal to each other: L{B) = ¥@EL(B)

(1.1)

(Corollary to Theorem 7.4 in the Appendix). We denote by F the orthogonal projection of L{B) into F. Then there exists a linear map G of L(B) to EL(B) such that, for any il/eL(B), EGij/ = GEIIJ = IIJ-FII/

(7.8)

(Theorem 7.4 in the Appendix). The linear map G is called the Green operator. Let A G C be a complex number. If there exists ee L{B) with e9^0 such that Ee = Xe, then we call A an eigenvalue of E and e an eigenfunction of E. From the assumption that E is formally self-adjoint, it immediately follows that all the eigenvalues of E are real. As regards the eigenvalues and eigenfunctions of E, the following theorem is of fundamental importance. Theorem 7.1. We can choose eigenfunctions e^ e L{B) of E with Ech = XhCh, /i = 1, 2 , . . . , satisfying the following conditions: (i)

{ei, e2,..., Ch,...} form a complete orthonormal system of L(B). Namely, (Ch, e^) = 6^,^, and any if/e L{B) is expanded into the series oo

'/'= E {^,eh)e,

(7.9)

h= l

(ii)

which converges with respect to the norm \\ ||. A i ^ A 2 ^ - • - ^ A ^ ^ - • • , and Xh^-^^ih-^-^00).

Proof of Theorem 7.1 will be given in the Appendix, §7(b). Since (Eij/, Ch) = ((A? ^^H) = ^h(^, ^h), we have oo

Eilj= I A,((A,e,)e,.

(7.10)

h=i

Suppose that • • • ^ A^ < A^+i = Xq+2 = - • • = A^ = 0 < A^+i ^ • • • . {Cq+i,..., Cp} form an orthonormal basis of F. Therefore, we have F^=

I

i^,e,)e,.

Then

(7.11)

324

7. Theorem of Stability

As for the Green operator, we have GilJ= I

-^(^,e,,)e,.

(7.12)

In fact, since Gif/ e EL(B), we have FGif/ = 0. Therefore G(A= I

{Gil/,eh)eh.

On the other hand, since EGil/ = ij/- Fif/ by (7.8), we have

Hence we obtain (Gi//, e^) = (l/Xh)(^, ^h), and (7.12) is proved. (b) Differentiable Family of Partial Differential Operators In this subsection, we consider a family of partial differential operators Et acting on L{Bt), where Bt is a complex vector bundle over X depending on the parameter t. We suppose that the parameter t moves in a small domain A of R^. We first define the notion of a family {Bj ^ G A} of complex vector bundles over X being differentiable in t Let ^ be a complex vector bundle over X x A and TT: ^ ^ X x A its projection. Then for each r G A, B, = 7r~^{X x r) is a complex vector bundle on X = X xt. Definition 7.3. A family { B J ^ e A} of complex vector bundles Bf over X obtained as above from a complex vector bundle B over X x A is called a differentiable family of complex vector bundles over X. Since we only consider the case in which A is sufficiently small, choosing an open covering { Uj} by sufficiently small coordinate neighbourhoods Uj, we may assume 7r-H^-) = C ^ x ^ . x A . The points ({y, x, r) e C x ^- x A and ( 4 , x,t)eC'' point of ^ if and only if ^j= i

h%^{x,t)CL

xUkX^

are the same

(7.13)

/x = l

where bjk^(x, t) is a C°° function on (L/^ n L4) x A. For each member Bt of the differentiable family {JBJ ^ e A}, we use ibjj^^(x, t)) as transition functions.

§7.1. Differentiable Family of Strongly Elliptic Differential Opbrators

325

We also use the fibre coordinates ( ^ j , . . . , (J) of^ as those of each Bf. These fibre coordinates on Bt are called admissible fibre coordinates of Bf. F8r example, if we say that a C°° section i/^^ e L(Bt) of Bt is represented bj^ a vector-valued C°° function on Uj as (A,(X) = ( ( A ! , ( X ) , . . . , ( A O W ) ,

(7.14)

we always suppose that ((/rjy(x),..., il/yix)) are the admissible fibre coordinates of il/t{x)e Bt. Hence, on Ujn U^T^ 0 , we have ^tj(x)=

i

bf^^{x,t)ilj'tk{x).

(7.15)

Definition 7.4. Suppose we are given a C°° section (/r^ e L{Bt) of Bt for each re A. If each admissible fibre coordinate «A^(jc) of i/^^x) is a C°° function of (x, 0, then if/t is said to be C°° differentiable in t. Put (Aj'(^, 0 =

^?,(A:).

Then, by (7.15)

hence, ^: (x, t)->^j{x, t) = {IIJ]{X, t),..., il/J(x, t)) is a section of ^ over X x A , and each ij/t is the restriction of^ ip to X xt: il/t = ^\xxf In other words, to say that the if/t e L{Bt) are C°° differentiable in t is equivalent to that each if/t is the restriction to X x r of a C°° section if/ of ^ over X x A. Definition 7.5. Suppose we are given a linear operator A^: L{Bt) -> L(B^) for each re A. If A,(/^f is C°° differentiable in r whenever [fjt^L{Bt) is C°° differentiable in r, then A^ is called C°° differentiable in r, and the family { A J ^ G A} is called a differentiable family of linear operators. Suppose that a linear differentiable operator £,: L{Bt)-^ L{Bt) is given for each r e A. By (7.1), for arbitrary (/^, G L{Bt), we have iEtilft)fM = i

El(x,

t, D,)eA^(x).

(7.16)

Put 1^7 (x, t) = il/^j{x), (pj(x, t) = (p^j{x) where (Pt = Et^t- Then (7.16) is written in the form

9t(x,o= i £j;.(x,r,D,)^;(x,o. The C°° differentiability of Et in r means that, if the iff'^ix, t), IJL = \, ..., v arc C°° functions of (x, t), then the
326

7. Theorem of Stability

functions of (x, t). Therefore, E^ is C°° differentiable in t if and only if the coefficients of the polynomials Ej^{x, t, Dj) in Dj^,..., Dj^, A, /x = 1 , . . . , i^, are all C°^ functions of (x, t). Assume that a Hermitian metric on the fibre Tal^=i ^jx/ii^, t)^j^f is given on ^ . Then we define the inner product of if/t, (Pt ^ L{Bt) by I aj,4x,t)ilj^j{x)
i^t,(pt)t =

(7.17)

X A,)u,

It is important to note that the aj^^iix, t) are C^ functions of {x, t). Hereafter, when we say a linear differential operator E^ on L{Bt) is formally self-adjoint or strongly elliptic, it is so with respect to the inner product ( , )t defined by (7.17). Suppose we are given a differentiable family {Et\teL} of formally self-adjoint strongly elliptic linear partial differential operators E^ on L{Bt). By virtue of Theorem 7.1, we determine, for each Et, the eigenvalues kh{t), and eigenfunctions Cth e L{Bt) with EfCth = Xh{t)eth satisfying the following conditions: (i) (ii)

{eth\h = 1,2,...} form a complete orthonormal system of L(Bt). A i ( 0 ^ - • • ^ A / ( 0 ^ - • •, and Xhit)-^oo (h^oo).

Let ¥t = {il/e L(Bt)\Etijj = 0}, and let F^ be the orthogonal projection of L(5,) toF,. Then by (7.11), Ft^=

Z

(^,eth)eth,

ipeL{B,).

AA(0=O

In this section we shall prove the following three theorems ([21]). Theorem 7.2. kh{t) is a continuous function

ofteA.

Theorem 7.3. dim F, is upper-semicontinuous in ^ G A. Theorem 7.4. If dim F, 15 independent ofteA, teA.

then Ft is C°° differentiable in

Note that Xh{t) is not necessarily differentiable in t. This c a n b e seen, for example, by the matrix (} {) whose eigenvalues Ai(0 = 1+V^ A2(0 = are not differentiable at ^ = 0. We give below proofs of Theorems 7.2, 7.3, and 7.4. Obviously we may assume that the domain A is a multi-interval: A^={teU^\-s
= l,..., N},

e>0.

Furthermore, if necessary, we may replace Ag by a smaller A^, with 0<e' and prove the theorems for A = A^.

<e,

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

327

Recall that each vector bundle Bt in a differentiable family {Bt\teA^} is the restriction to X x ^ of the vector bundle ^ over X xA^: Bt = ^\xxt = 7T~^{X X t), where TT is the projection of ^ . On the other hand, it is obvious that Bo X Ag is a vector bundle over X x A^. Lemma 7.1. The vector bundle ^ over X x A^ is C°° equivalent to the vector bundle

BQXA^.

Proof. Let TTQ: BQXA^-^X XA^ be the projection of BoXA^. We want to construct a diffeomorphism ^ of the JBQ x A^ onto ^ which maps each fibre 7ro\x, t) linearly onto 7r~\x, t). For this, we apply a method similar to the proof of Theorem 2.4 (pp. 64-66). Let {Uj} be the finite covering of X as mentioned above, and put % = 7r-\Uj XAJ = C" X UjXA,. Then ^ = U j %, where ({,, x, 0 G % and (^fc, x, t) e ^ ^ are the same point on ^ if and only if ^j=lb%^ix,t)C^. (1°) The case N = 1. We denote by (d/dti)^ the vector field d/dt^ on ^fc. Let {pfc(x)} be a partition of unity subordinate to the covering {L4}. Then v = lPk(x)l

—)

k

\dti/k

is a C°° vector field on m. On % n^j,9^0,

we have

In view of CZ = 1^ Kj^(x, ^i)^f, if we set vfM

h) = I P.(x) I ^^^^-/^^- ''^ bZjAx, t,\ k

o-

oil

then t^jju,(x, ^i) is a C°° function on t/j x A^, and we have - p i U ' , ) & ! : + ( £

CIS)

on %;. Note that (7.18) is similar to (2.25) except that here the coefficients 1^ < ( x , t,)C^ of d/d^}. are linear in ij, ...,CJ.

328

7. Theorem of Stability

Consider the simultaneous ordinary differential equations for v: ^ =

i

vlix,t,)cr,

A = l,...,n, (7.19)

dt For each point (^o> x, 0) of Bo= 7r"'(X xO), we consider the initial conditions [^'(0) = ^o,,

A = l,...,n,

|<.(0) = 0, and let \0it) = cH^oj,x,t),

\ =

i,...,p,

be the solutions of (7.19) satisfying these initial conditions. Then ^I': U^j, X, t) -* Uj,x, /,) = UjUoj, X, t), X, t) is a diffeomorphism of BQXA^ onto ^ . Since the equations (7.19) are linear in Cj, • • •, ^j, the solutions i^Uoj, x, t)

Hence ^ maps each fibre TTOH^, 0 of ^o^-^e linearly onto 7r~^{x, t). (2°) General case. We can prove the general case by induction on TV just as in the proof of Theorem 2.4. I The diffeomorphism ^ : BoXA^^ ^ is written on each 7TO\UJ X A ^ ) as

^ : Uoj.x, t)^Uj,x, t),

Cj=lfjM

t)ar

In other words, ^ is a vector bundle obtained from BQ x A^ by the change of coordinates {^oj, • • •, Coj) ^ (^j, • • •, ^/)- Therefore ^ is the same vector bundle as BQxA^, and each Bt in the differentiable family {Bt\te A} is the same as BQ. Consequently, we may assume that the Bt = B are one and the same vector bundle independent of t, and consider a differentiable family {Et\te^} of partial differential operators Et on L(B).

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

329

The Hermitian metric on the fibre Y^x^ ^j\fi(^, ^)C^^f of the vector bundle ^ = B X Ag over X x A^ depends in general on t. Therefore the inner product given by (7.17): (^,
I a,,^{x, t)^]{x)ipf{x)J~g

dX,

il;,cpe L{B)

J X \,fJL

also depends on t. Assume that each E^ of the differentiable family {Et | ^ e A} is a strongly elliptic linear partial differential operator of even order m. Then, if we write Et in the form (Etilj)f(x)=l

Z

Elix,t,Dj)iljr(x),

then the equality (7.4): I

aj^^{x,t)EJl-{x,t,Dj)=

o-=l

I

aj,^(x,t)ET^-{x,t,Dj)

o-=l

holds, and if we put

X,IJL,cr

the inequality (7.6):

A(x, t, i, a ^ 8'\ir I a.^ix, o^'r,

5> 0,

(7.20)

holds. Restricting the domain of t to A = Ag' with e ' < e , so that [A]c: A^, we may assume that 8 is a constant independent of t. We put Dj^ = d/dxj as before. We let D] denote an arbitrary differential of rank /:

We define the norm ||jA||fc, /c = 0, 1 , . . . , of i/^G L{B) by

\l = 0 j

Dj J Uj ^

I

where the summation X^' is taken over all partial differentials D] of rank /. The norm \^\k is equivalent to the Sobolev norm given by (3.7) in the Appendix. Namely, if we denote the Sobolev norm by || ||/f^\ then, by Proposition 1.2 in the Appendix, there is a constant K > 1 depending only

330

7. Theorem of Stability

on k such that

In particular, for A: = 0, we have

0=1

II
[

dXj,

j J Uj A

while as for the norm \\if/\\t = ^{if/, i(/)t, we have IIf{x)/g

dX.

J X A,/x

Therefore there is a constant Ko> \ independent of ^e A such that I.^Koll^llo

and Mo^KoM,

(7.21)

Lemma 7.2 (Sobolev's Inequality). For any integer / ^ O , and any natural number k>n/2 with n = dimX, there exists a constant c^i such that the inequality \D]il,^{x)\^Cu,M\k^u

il^eL(B)

(7.22)

holds. Proof. See (3.33) in the Appendix. I Lemma 7.3 (Friedrichs' Inequality ([6]). Let k be a non-negative integer. Then there exists a constant c^ independent of t such that the inequality (UU^J'^Cj^QEMll^Ml).

il^eL(B)

(7.23),

holds. Proof Since {Et 11 e A^} is a differentiable family of strongly elliptic operators and since [A]c: A^, by Theorem 4.1 in the Appendix, there is a constant Q such that (UU^^r^C^(\\EM\l-^Ml),

^eL{B).

(7.24)

If k
§7.1. Differentiable Family of Strongly Elliptic Differential Operators

331

(7.23)fc. By (7.24)

while by (7.23)fc-i

Since HEfiAUfc^ig ||£,
(Uh^J'sail

+ c.KWEMi+Ml)-

Hence, putting c^ = Cfc(cfc_i + 1), we obtain (7.23)^.

I

Theorem 7.5. Assume that Et'. L{B) -» L{B) is bijective for each teA. If there exists a constant c>0 independent of te^, such that, for any if/e L(B) \\E,il,\\o^cMo,

(7.25)

then E~^^ is C°° differentiable in ^G A. Proof First we extend Definition 7.4, and define the notion that il/tE L(B) is C^ differentiable in ^G A. On each Uj, we represent if/t as ilj, =

{ilj]{x,t),.,,,iljj(x,t))

in terms of the fibre coordinates. If the D]IIJJ(X, t), A = 1 , . . . , ^, are C functions of (x, t) for any Dj, then t/^^ G L{B) is called C^ differentiable in ^GA. Furthermore, a linear operator At'. L(B)-^ L{B) is said to be C^ differentiable in t, if, for any I/^^G L(B), C ^ differentiable in t, Kt^t is also C^ differentiable in t. In order to show that E~^^ is C°° differentiable in r, it suffices to prove that £7^ is C^ differentiable in t for each r = 0, 1, 2, By hypothesis, we have, for t e A, ||iA||o^-||£.^||o^-||^.^IU, c c

iljeL(B),

Hence by (7.23)^

where Ck = c]/^(\-\rl/c) is a constant independent of ^ G A and if/. On the other hand, by Sobolev's inequality, we have for k-\-m — l> n/2, \D'jll^fix)\^C^^m-l,lMk^m^

332

7. Theorem of Stability

Hence, for k> l — m + n/l, the inequality |DJiA,'(x)|^4,,||£,(AlU

(7.26)

holds for any if/e L(B), where Ck,i= c'j^ * Ck+m-ij is a constant independent of t. We prove, by induction on r, that, if ^^ G L(B) is C^ difierentiable in t, so is ilJt = EJ^(pt. (1°) The case r = 0. By hypothesis (pt = Etil/t is C^ difierentiable, i.e., continuous in re A. We need to prove that for any Dj, D]il/j{x,t) is continuous in (jc, t). Since Djipfix, t) is continuous in x, it suffices to show that, for any 5 G A, D]ip^{x, t) converges to D]II/J(X, S) uniformly in XG UJ as t-^ s. For given /, we choose k with k> l—m-\-n/2 once for all and put c' = c'kj. Then, by (7.26)

|DJ(At(^,0-^>,'(^,5)Nc'||£,((A.-^s)IU = c'||(p,-9,|U + c1|£,(A.-£AlU.

(7.27)

On the other hand k

\D^jcp^(x,t)-D^j
\
where the second X designates the summation over j , D] and A. By hypothesis D](p^{x, t) is continuous in (x, t). Hence \^(pt-(ps\\k-^^{t-^s). For fixed s, m

(£,^j;(x)=I

V

I £jt(^, r, D,).//S.(x)

is a C°° function of (x, 0 because, in the right-hand side, the Dj\ ' • • Dj'}iil/^j{x) are C°° functions of x, and their coefficients are C°° functions of (x, 0- Hence \\Etilj,-E,ilj,\\k-^0 (t-^ s). Therefore, by (7.27), as t-> s, we have D]II/J(X, t) -^ D]{1/J(X, S) uniformly in x G UJ. (2°) The case r = 1. Put (pt = Etij/f We must show that if cpt is C^ differentiable in t, so is i/jf We denote by ( f i , . . . , ^TV) the coordinates of r G R^. Since by hypothesis cpt is C^ difierentiable in t, each component (pf{x, t) of <^f = (
It is obvious that dcpt/dt^ G L(B).

d(pj(x,

t)\

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

333

Suppose il/t is proved to be C^ differentiable in f G A. Then, differentiating the equation (pt = Etif/t with respect t^, we get

— = Et

1- — il/t,

hence,

'l=E;i^-^^X

dt,

(7.28)

where dEJdt^ is the partial differential operator obtained from Et by differentiating with respect to t^ the coefficients of Ef^{x, t, Dj) as polynomials of Dja, a = 1 , . . . , m. Namely, for any ij/e L(B),

(

riE

\^

^

fi

T7
(7.29)

By hypothesis d(pt/dt^ is continuous in teA. Since cpt is continuous in re A, 4ft = E~^^(pt is continuous in r by (1°). Hence r]^t is continuous in t, that is, for any D], D]T)^^J{X, t) is a continuous function of (x, t). Hence, in order to prove that (/^^ is C ^ differentiable in t, it suffices to prove that, for any DJ, Djij/jix, t) is differentiable in r^, K = 1 , . . . , N, and that

That is, putting t + h = {ti,...,

r^_i, t^ + h, t^+u • • •, ^iv),

we need to prove lim^(DJ(A'(x, f + / i ) - D > ' ( x , 0 ) - i ^ j ^ K A 0 = 0. For this, by (7.26), it suffices to show lim uB,+4 -ri^t^-h - ^t) - VKI I = 0. ^^o|| \h I

(7.31)

334

7. Theorem of Stability

Using the equalities Et+h^t+h = (Pt-^h, E,ilj, =
-
dcpt

1 --(Et+h~Et)il/t n

dE +

—^il/t-(Et+h-Et)r],t.

dt^

Therefore, in order to prove (7.31), it is enough to prove lim

lim

1/

^^t\

X

= 0,

(7.32)

-0.

(7.33)

Et)r]^t\\k=0.

(7.34)

-{Et+h-Et)il/t-—^il/t h dt^ lim\\(Et+h-

From the definition of the norm || H^, in order to prove (7.32), we have to show that, for any Dj, I ^ fc, t)

dXj = 0

holds. But this is obvious. In fact, since (pt is C^ difierentiable in t, and since the fibre coordinates (fj , . . . , ^J) of B are defined on some domain ^j^mi it follows that, for a sufficiently small 5 > 0 , the D](pj(x, t + h) are continuously difierentiable functions of ( x j , . . . , xj, h) in Wj x ( - 5 , 5), with t being fixed. Therefore lim ^'•^^^^'^- t-hh)~D]
h

t)

dt^

uniformly in XG UJ. This proves (7.32). Put m

f^{x,t

+ h) = (E,^Mf{x)=

I ):E'i'^{x,t + 1=0

h,DjHtjix).

IX

For fixed r, the/;^(x, t-^h) are C°° functions of ( x j , . . . , xj*, h) in W; x ( - 6 , 8) because the coefficients of Ef^{x, t, Dj) as polynomials of D , i , . . . , D^„ are C°° functions of (x, 0- In order to show (7.33), it is enough to prove dXj = 0.

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

335

This is also obvious. For, since by (7.29)

we have h->0

h

-/\ -..

^r /

V

/

uniformly in x G UJ. (7.34) can be proved similarly. (3°) The case r ^ 2 . Put (pt = E^ilft as before. We shall prove that if (pt is C^ differentiable in ^ e A, so is e/^^. By the hypothesis of induction, \pt is C'^ differentiable in t. We denote a partial differential operator of order k in ^ 1 , . . . , r^ by

dt\'' •' dty Since

dr\E,il/,) is a sum of E.dr^ip, and the terms of the form (ar'"''J5,)af iA„ A: = 0,l,2,...,r-2:

Hence

fc = 0

Since (^, is C differentiable, if/t is C " ^ differentiable and Ef is C°° differentiable in t, it follows that the right-hand side of the equality is C^ differentiable in t. Hence, by (2°), d\~^^t is C^ differentiable in t. This implies that i/^, is C^ differentiable in t. This completes the induction. I In order to prove Theorems 7.2, 7.3, and 7.4, we assume in the following that each member Et of the differentiable family { E j r e A } is formally self-adjoint and strongly elliptic with respect to the inner product ( , ),. Suppose we have chosen the eigenvalues A^(0 and eigenfunctions CthE L{N) with EtCth = ^h(t)eth which satisfy the following conditions: (i)

{Cth I /i = 1,2,,..} form a complete orthonormal system of L(B) with respect to the inner product ( , )t.

336

7. Theorem of Stability

(ii) A,(0^A2(r)^"-^A„(r)^---, For a moment, we fix teA = A^', 0<s'<s. expanded into a series

A,(0->^

(h^oo).

Any element il/eL{B)

is

which converges with respect to the norm || ||^ In this way, il/eLiB) represented by the sequence {ah}. Clearly we have

is

oo h= \

i \an\'=\ml

(7.35)

h=l

Lemma 7.4. A necessary and sufficient condition for a sequence {ah} to represent a element ofL(B), that is, a condition for the existence of if/ e L{B) satisfying ^ = YJh=\ ^h^th'> is that the inequalities I |A,(Ona,|^<+oo,

/=1,2,....

(7.36)

should hold. Proof The necessity of (7.36) is easily verified as follows. If t/f = X?=i ^/i^rh e L{B), then E\il/eL{B). On the other hand, since by (7.10), E[il/ = Ih^hitYahCth, we have

i iA,(ork.r=ii^;(Aii?<+oo. h= \

To prove the sufficiency of (7.36), we first show that, for any ij/ = T.°h=i ^h^th e L{B) and any /c= 1 , 2 , . . . ,

UWUc'L Z ( l + i \kh{t)\A\ah\'

{131)

holds, where c'i is a constant independent of i//. Since the inequality (7.23)^:

{m^^^f^c,{u\\inE^Hi) remains true if we replace c^ by a larger constant, we may assume that Cfc ^ 1. Then we have

l^co{U\\l+\\EM\l\

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

337

Similarly, for ^ = 3, 4, 5 , . . . , we have

nCo(iiiAiio+iji£;^iio).

2 < ^ qm — ^(q—l)m

For any given natural number k, we choose q so that qm-m
i^mu Therefore, in view of (7.21) || ||o=i^o|| \\h we obtain

^ 4-Ko( 11^11?+IJlsJ^II?). Since |?= Z | a , r

and

||£',.A||?= Z

\^uitr\a,\\

(7.37) immediately follows from this inequality if we set c^ = c^Kl. Suppose we are given a sequence {a^,} satisfying condition (7.36). Since ^hit)-^ + ^ as /z ^ 00, it follows immediately from (7.36) that Xr= i I'^/i 1^ < + ^ Hence, we have i

( l + i |A,(Op)|a.|'<+oo,

h=\ \

1=1

^=1,2,....

(7.38)

J

Put t/^^^^ = YX=\ ^h^th' Of course, we have if/^^^ e L{B). In order to prove the existence of if/e L{B) with lA — Zr=i ^h^th, it suffices to show that for each Dj, I ^ 0, the sequence {D]il/^/^^{x) | ^ = 1, 2,...} converges uniformly on Uj. Needless to say, D^(/^j^^'^(x) = i/^j^^^(x). For any pair of natural numbers p, q with p l+n/2, we obtain from Sobolev's inequality (7.22) and the inequality (7.37)

^(Cfc-ufci:

i h=p+l

( l + E |A,(Or'')|a,p. \

(x = l

J

By (7.38) the right-hand side converges to 0 as ;? -> oo and ^ ^ oo. This proves that {D]^Y^^{X)'] converges uniformly on JJj. I

338

7. Theorem of Stability

Let C t>e an arbitrary complex number and put

Then EfU) is a strongly elliptic differential operator acting on L{B). For any
i

i\,{t)-OaHe,,.

(7.39)

h= l

If f 7^ A^(r), /i - 1, 2 , . . . , then £,(^): L{B)-^L{B) is bijective. Proof. It is clear from (7.39) that EtU) is injective. In order to prove that E^O is surjective, take any (p e L{B) with (p = ^^=1 h^th, and put Uh = hh/{kh{t) - ^). By Lemma 7.4, we have

i iA,(onb,p<+oo,

/=i,2,....

Therefore, since Ah(r)7^f and Aft(0^+oo as /i-»oo, we have

Hence, again by Lemma 7.4, there exists if/e L{B) with lA — Zr=i ^^^?^- We obviously have E^O^^ 0, there exists a constant c > 0 such that, for \t — to\<8 and \( — ^o\<8, the following inequality WEXn^o^cMo,

(AeL(B)

(7.40)

holds. Proof. Suppose that, for any small 6 > 0, there is no such constant. Then, for ^ = 1, 2 , . . . , there exist r^ G A, ^^ e C and (/^^^^ e L{B) such that

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

339

By (7.23)o

Hence, in the equation

we have (*

and the coefficients Ej^i^...i^{x, t) are C°° functions of (x, 0- Hence

Since ||£,^(f,)iA^^^||o
On the other hand, by (7.39), we have for any ipe L{B) ||^/6(^o)'AlU = Mo||'AlU,

^o = min|A^(ro)-fol>0.

Hence, by (7.21) Kl\\EtJ^Co)^h = l^oW^^o- Consequently we have ||oo), which contradicts ||tA^^^I|o= 1- I Suppose we are given ^o^ ^^ and Co"^ ^h{to), h = l,2, If we take a sufficiently small 5 > 0, then, by this lemma, ||Ef(^)i/^||o= c||(/^||ofor|f — ^o| <^ and 1^- ^ol < ^, hence, it follows that ^ 9^ A^(0, h = l,2, Therefore EtiO is bijective, and, by Theorem 7.5, G^(^) = Et(^)~^ is C°° differentiable in (r,^). If we put

w={(t,ae^xc\^^x,{t),h

= l,2,...h

then, for any point {to, ^o) ^ ^^^ we can find a constant 5 > 0 as above. This implies that W is an open subset o/ A xC, and Gt{C) is G°° differentiable in

(UnonW. Fix an arbitrary ^o ^ A, and take a closed Jordan curve C on the complex plane C which does not pass through any of the \h{to),h = l,2, The Jordan curve C divides the complex plane into two domains, the interior and the exterior of C. We denote by ((C)) the interior of C. Since

340

7. Theorem of Stability

Figure 1

C xto^ W, and since W is open in A xC, we have Cx[to-8,to+8]c:W,

(7.41)

provided that 5 > 0 is sufficiently small. For any t with \t-to\<8, the linear operator F ^ C ) of L{B) by

we define

A;,(OG((C))

and put Ff(C) = Ft{C)L{B). Obviously, Ff(C) is a finite-dimensional linear subspace of L(B), and it consists of all the linear combinations of those eth which belong to the eigenvalues A^(r) lying in the interior of the Jordan curve C The Hnear operator Ft{C) is nothing but the orthogonal projection of L(B) onto F^(C) with respect to the inner product ( , )t. The operator F^C) can be written as follows, using the Green operator GACY Om^dC, ^-Mc""'

F,{C)4^ = - - - \

iljeLiB).

(7.42)

Proof. Since, by (7.41), Cx[to-8, to-\-8]c:W, and since G,{0 is C^ differentiable in (t, 0 in "W, the integral l^^tiO^dC exists. We let i/^ = S^^^ a^e,^. Then, since G,(^) = E(ty\ we obtain by (7.39)

°°

au

Hence, by Cauchy's integral formula,

^ , . G,{0^dC= 2771

Jc

I

a,e,, = F,(C)iA.

A,(OG((C))

Lemma 7.6. Ft(C) is C°° differentiahle in t for

\t-tQ\<8.

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

341

Proof. Since G,(^) is C°° differentiable in {uO in W, it follows that, if iptELiB) is C°° differentiable in t for \t-to\<8, then F,(C)iA, = -(1/277/) Jc GtU) 4ft dC is also C°° differentiable in r. I Lemma 7.7. dim Ff(C) i5 independent of t for \t — tQ\<8 provided 5 > 0 is sufficiently small: dim Ff(C) = dimFf^CC). Proof Put dim F^(C) = d, and let {e^,..., e^,..., e^} be a basis of F^(C). By Lemma 7.6, the FXC)^^ are C°° differentiable in r, and the FtJ^C)er = e„ r = 1 , . . . , J, are linearly independent. Therefore, the F,(C)e^GF^(C), r = 1,... ,d, are linearly independent for | r - ^o| < 8 provided 5 > 0 is sufficiently small. Hence d i m F ^ ( C ) ^ d Suppose that for any small 6 > 0 , there exists t with \t-to\<8 such that dimFf(C)> t/. Then we can find a sequence tq, q = 1,2,... ,with\t-to\m + H - n / 2 , then, by (7.22) and (7.37), for each DJ, / ^ m + 1,

Since xi'^^e ({€)), the sequence {Djeif''{x)\q = l,2,...} is uniformly bounded in Uj. It follows that, for any / ^ m, {Dje^/^'^(x)} is equicontinuous on Uj. Hence, we can find a uniformly convergent subsequence. Taking an appropriate subsequence of {t^}, we may assume, from the beginning, that {Djcif^ix)} converges uniformly in Uj for any £>j, l^m. Put e^j{x) = limg_^oo eif^(x). Then e^j(x) is a C " differentiable function on Uj, and we have lim D^jcifix)

= DJ4(x),

/ ^ m.

Therefore the e^ = lim^^oo ^r^^ are C"" sections of B and e^j(x) are the values of their fibre coordinates. From (ei'^\ e^^^)^^ = d^s, it follows that

Since the Ef^{x, t, Dj) are linear combinations of D] with / ^ m, with the coefficients being C°° functions of (x, r), we have limF,^e(.^> = F^e..

342

7. Theorem of Stability

On the other hand, we have £f^et^^ = A^^^e^^^^ Therefore, the Umit A^ = Um^^oo A^^^ exists, and we have

This means that A^ is an eigenvalue of f"^, and e^ is an eigenfunction belonging to A^ Since A^-^^G ((C)), and since there are no eigenvalues of £^ on C, A^ must coincide with one of Xh(to) &{{€)). Hence e^GF^(C), r = 1,2, . . . , d + l. But, since (^n ^s)to = ^rs, it follows that ^ i , . . . , e^+i are linearly independent. This contradicts dim F^(C) = t/. I Proof of Theorem 7.2. We shall prove by induction on h that lim^^^ A^(0 = A;,(ro) for roGA. (1°) The case h = l. By (7.14) in the Appendix, there is a constant j8 independent of t such that Ai(0>j8 for teA. Taking a sufficiently small e > 0, we consider the circle C^ of radius s with centre Ai(^o) and a smooth Jordan curve C which intersects transversally with the real axis at (3 and A = Ai(ro) - e. We assume that C does not intersect with the real axis except for at /3 and A. Take a sufficiently small 8>0. Since there are no eigenvalues kh(to) of ^to i^ the interior of C or on C, we have dimF^(C) = 0. Hence, by Lemma 7.7, dim ¥t(C) = OfoT\t~to\< 8. Therefore, Ai(0 is a real number outside of C. But, since \i{t)> p, we have A i ( 0 > A =\i{to)-e.

Figure 2

Moreover, by Lemma 7.7, we have dimF,(Ce) = dimF^(Cg)^ 1 for \tto\<8. Hence there is at least one eigenvalue Xh(t) in the interior of Q . This implies that \^(t)^\h(t)<^i{to)-^ e. Therefore |Ai(r)-Ai(ro)|< £ holds if |f - rd < 8. This proves that lim^^^ \i(t) = \i{to). (2°) The case h^2. We assume that lim^^^ Afc(0 = Afc(^o) for k = 1,2, . . . , / i - l , and prove lim,_,^ A;,(0 = A;,(ro). If Ai(ro) = • • • = Xhito), then, for the circle C^ of (1°), we have, by Lemma 7.7, dim F,(Ce) ^ /i for |^ - ^ol < 5, provided 5 > P is sufficiently small. Therefore Ai(0, • • . , ^hit) lie in the interior of C^. Hence \^h(t) -\hito)\ < ^- This proves lim,^^ A;,(0 = A^(ro).

§7.1. Differentiable Family of Strongly Elliptic Differential Operators

343

Next we let / be the integer, 2 ^ / ^ /i, such that

Ai(0 ^ A/_i(ro) < A , ( 0 = A,+i(ro) = • • • = A,(ro). Let Che he the circle of radius s with centre \h(to), where e is assumed to satisfy 0 < e < A/(fo)-A/_i(fo), and let Q be a closed Jordan curve which intersects transversally with the real axis only at the two points \= \i{to)-e and fjL = \h(to)~^' We take 8>0 sufficiently small depending on Che and Ch. Then, since at least / i - Z + l eigenvalues Xi(to),..., \hito) are in the

Figure 3

interior of Che, we have d i m F ^ ( Q g ) ^ / i - / + l . Therefore, by Lemma 7.7, at least h-l+l eigenvalues A^CO's are in the interior of Che- On the other hand, from dim F^(C,,) = / - 1 , it follows that dim F,(C,,) = / - 1 for | r - ^ol < 8. Hence, exactly / - I eigenvalues A^CO's are in the interior of Q . By the hypothesis of induction, lim^^^ Afc(0 = Afc(^o) for fc = 1, 2 , . . . , / - L Therefore Ai(0, • •., A/_i(0 are in the interior of Ch for \t- tol < 8. This implies that A / ( 0 , . . . , A;,(0, • •. are all outside of Ch- Hence,

Combined with what we have seen above, it follows that A/(0, A / + i ( 0 , . . . , A^(0 must lie inside of Che- Hence |A;,(0~A;,(^o)|< ^ for \tto\ < 8. This proves lim,_^^ A/,(0 = A^(ro). I Proof ojf Theorem 7.3. By definition, ¥t = {ipe L{B)\Ftilj = 0}. We need to prove that, for a given toe A, we can find a sufficiently small 8>0 such that dim ¥t = dini F^ for | r - ^ol < 8. If we let Q denote the circle of radius e>0 with centre 0 in the complex plane C, then F^ = F^(Ce), provided e is sufficiently small. Therefore, by Lemma 7.7, dim ¥t(Ce) = dim F^ for |r - ^ol < 8, where 8 is supposed to be sufficiently small. Since the inclusion F^ c: F^(Q) is obvious, we obtain dim F^ ^ dim F,(Ce) = dim F^^. I Proof of Theorem lA. By diefinition F^ is the orthogonal projection of L{B) onto F, with respect to the inner product ( , ),. In general, for any toe A,

7. Theorem of Stability

344

if we take a sufficiently small 8, then, as we have seen above, the inclusion FfCzF^Cg) holds for \t-to\<8. Since we have dimF,(Q) = dimF^, and since dimF^ is independent of t by assumption, we conclude F^ = Ff(Ce). Hence Ft = Ft{C^). But by Lemma 7.6, F ^ Q ) is C°° differentiable in t for I r - ^ol < 5. Since ^o is an arbitrary point of A, it follows that F, is C°° differentiable in r G A. I Let Gt be the Green operator of Ef. Then, by (7.12), we have, for

G4=

I

(7.43)

-y-e,,.

Theorem 7.6. If dimF^ is independent of te^, in t G A.

then Gt is C^ differentiable

Proof We prove this by a method similar to the proof of Theorem 7.4. For any ^o ^ ^, we take a small circle C with centre 0 such that all the eigenvalues A / , ( O ^ 0 are outside of C. If 5 > 0 is taken sufficiently small, then, by (7.41), C x[^o- 5, ^0+ 5 ] ^ '^. For t satisfying \t-to\<8, we define the linear operator Gt{C) on L{B) by the formula

-h\>^

-Gtin^dC

Gt(C)ilf = --,

Since G,(^) is C°° differentiable in (r, C) in W, it follows that G,(C) is C" differentiable in t for \t-tQ\<8. For any YX=\ ^h^th, we have oo

^

By the residue theorem, 1

\uit)eiiC))..

Ini J, Hence for any JA ==Zr=i ^^^rh, we have

Since, by hypothesis, dim F^ is independent of t, F, = Ff(C) for | r - fol < 5 as was proved in the proof of Theorem 7.4. Therefore, the equality A^( r) = 0

§7.2. Differentiable Family of Compact Complex Manifolds

345

amounts to the same as A^(0 e ((C)). By comparing (7.43) and (7.44), we obtain Gt = Gt{C). Consequently Gt is C°° differentiable in t provided \t-to\<8. Since ^o is arbitrary, Gt is C°° differentiable in rG A. I

§7.2. Differentiable Family of Compact Complex Manifolds In this section we prove Theorem 4.1, Theorem 4.4, and other results by applying the results of the preceding section.

(a) Differentiable Family of Compact Complex Manifolds Consider a differentiable family M = (M, I, m) = {Mt\te 1} of compact complex manifolds over a multi-interval / with O G / C : | R " (see Definition 4.1), where Mt = vT''^(t). By Theorem 2.3, the underlying differentiable manifold X of M^ does not depend on t, and by Theorem 2.4, there is a diffeomorphism ^: X xI-^M of X xl onto M such that m • '^ coincides with the projection X xl^ I. We identify X xl with Ji via ^ , and consider M = XxL We denote a point of X x / by (x, t). By definition, there are a locally finite open covering {%\j = \,2,.,.} of X x / and complex-valued C°° functions z](x, 0? • • • ? ^]{^-> 0 on ^j such that for each t, {x -> (z](jc, 0 , . . . , z]{x, t))\%nXxty^

0}

forms a system of local complex coordinates on M^. If we write z^(x, r) = (ZJ(A:, t),..., zj(x, t)), the map (x, 0 ^ (2)(x, 0 , 0 is a diffeomorphism of ^^ into C " x / and {{x,t)-^{zjix,t),t)\j

=

l,2,...}

is a system of local coordinates of the differentiable family Ji (see p. 185). On % n ^fc 7^ 0 , we have zf(x, t) =/ffc(zfc(x, 0, 0,

a = 1 , . . . , n,

where ffk{zk,t)=f%{z\,...,zl,t^,...,tj) are C°° functions of Zfc,..., Zfc, ti,... ,tm which are holomorphic in z j , , . . . , z^. Taking a sufficiently fine finite covering { ^ |j = 1, 2,...} of X, we may assume that for a sufficiently small multi-interval A = {te l\\t^\
346

7. Theorem of Stability

the map

(x, 0 -> {zjix, t), t) = (z](x, 0, •.., 2:;(x, r), ri,..., f^) gives local coordinates of the differentiable manifold J^A on Uj x A. Since the problem is local with respect to the parameter t, it suffices to consider M^ for a sufficiently small A. For simplicity, we write Jt = {M, A, m) for M^ = (M^, A, m) below. Let ^ be a complex vector bundle, and TT: ^^M its projection. Then Bt = 77~^(Mf) is a complex vector bundle over M,. We call such {B^ [teL] a differentiable family of complex vector bundles. In particular, if each Bt is a holomorphic vector bundle over the complex manifold M^, { B J ^G A} is called a differentiable family of holomorphic vector bundles. By saying that {Bt\te^} forms a differentiable family of vector bundles, we mean that {Bt\te^} is a differentiable family over the underlying differentiable manifold X in the sense of Definition 7.3. Since Uj and A are sufficiently small, we have 7r''(^xA) = C " x ^ x A , and a point of 7r"^( L^ x A) c: ^ is represented as {Cp x, t) = ( ^ ] , . . . , ^/, x, t) where ^j = (Z^j,..., ^J) are the fibre coordinates of ^ over Uj x A, and on UjXAn L 4 x A 7 ^ 0 , we have ^j=

i

h%^{x,t)a.

A = 1,...,K

Each component bjf^^{x, t) of the transition function bjk{x^ t) = (bfkfjiix, 0)A,A^ = I,...,I' of ^ is represented as a C°° function of the local coordinates (z^, t) = {zl{x, t),..., zl{x, t), t) of M by h%^{x, t) = bjk^izk, 0 ,

Zk = Zk{x, t).

Thus {Bt I ^ e A} is a differentiable family of holomorphic vector bundles if and only if all the bjk^(z,,, t) = bjk^{zi,..., z^, t) are holomorphic in z j , , . . . , zl, As stated before, we always use admissible fibre coordinates ( ^ ] , . . . , C]) for Bt. Let L{Bt) be the linear space of all C°° sections of Bt over M^. L{Bt) is nothing but the linear space of C°° sections of Bt over the underlying differentiable manifold X. We write a C°° section i/^^ G L{Bt) of B, by iA,(x, t) = (il/jizj, 0 , . . . , ^J(^,-, 0 ) ,

^j = ^i(^, 0

on UjXf^ M, where (
§7.2. Differentiable Family of Compact Complex Manifolds

347

and t. If each Bt is a holomorphic vector bundle over M^, and if/t is a holomorphic section of Bt, then ^A; (z,, t) is holomorphic in z, = {Zj,..., zj). Consider the tangent bundle T(Mt) of M,. The transition function of T{Mt) is given by

a,p = l,.

and its component df]l(zk, t)/dz^ is a C°° function of (z^, 0 which is holomorphic in Zj^. Consequently {T(Mt) | r G A} is a differentiable family of holomorphic vector bundles. If we write a C°° section ^^e L(T{Mt)) of T{Mt) as BzJ on Uj Xr, then we have on Uj xtn

fj (^> 0 - L

U^xt

7"^

^k{Zk, t).

Namely, (^](zj, 0 , • • •, ^J(^> 0 ) are admissible fibre coordinates. Therefore ^t is C^ differentiable with respect to ^G A if and only if ^"(Zj, t) are C°° functions of (Zj, t) for a = 1 , . . . , n, From (3.49) the transition function of the dual bundle T^'iMt) of T(M,) is given by

\

^Zj

/ cx,f3 = l,...,n

Consequently { r * ( M j | ^G A} is also a differentiable family of holomorphic vector bundles. A C°^ section (p^ e L{T''{Mt)) of T^'iMt) is a C°° (1, 0)-form on Mt'. n


(pjAzj,t)dzf.

a =l

{(Pjiizj, t),..., (pjnizj, t)) are clearly admissible fibre coordinates, hence cpt is C°° differentiable with respect to tG A if and only if each (PjuiZp t) is a C°° function of (z^, t) for a = 1 , . . . , n. z/ = z"(x, r) is a C°° function of (x, r) on Uj X A, and Jz" is the differential of zf as a function of x with t fixed. Thus, letting ( x ^ , . . . , x ^ , . . . , x^") be arbitrary coordinates of x, we have <=

I

'-^f^dx\

(7.45)

348

7. Theorem of Stability

where dzj{x, t)/dx^ are C°° functions of (x, t), which again shows that {(PjiiZj, t),..., (pjn(Zj, t)) are admissible fibre coordinates. Suppose given a differentiahle family ofholomorphic vector bundles {Bt \t e A}. Consider B,®/\'' T*(M,) A^ ?*(M,). Since {T*(M,)|rGA} is a differentiahle family,

{ - ® A ^*(M,)A

T*(M,)|rGA

is also a differentiahle family. A C°° section of B,® A'' '^*(M) A^ T'*(M,) is called a C°^ (/?, ^)-form with coefficients in Bf. Let ^^"^{Bt) be the linear space of C°° (/?, g)-forms with coefficients in B^:

(

p

9

_

i^^-'^CB,) = L B,® A T*(M,) A r*(M,) . (Pte£^^'^(Bt) is represented on each UjXtczMt ponents are C°° (/?, ^)-forms as

by a vector whose com-


(7.46)

where

1

^ j ^ j I ^ t , . . . ^ , ^ . . , - / z „ 0 dzf^ A . . . A J z ; . A J z f ^ A . . . A J z f .

On Uj®tnUkXt9^0,

we have

(^^ /5 called C°° differentiahle with respect to teA if the (pj^ ^ ^ ^ (z^, 0 ^''^ C°° functions of {Zj, t). In case Bt is a trivial line bundle C x M,) = LCA"" r*(M,) A^ T*(M,)) is the linear space of all C^ (/?, ^)-forms. dt(Pt is defined by a,<;p, = (dt(p]{Zj, t),...,

dt(pj(zj,

t)),

where d,
, i^J, t) dzf A dz> A dzp. P ^

§7.2. Differentiable Family of Compact Complex Manifolds

349

If (p, e <^^'^(B,), then we have a,
(Ot = ij, Pj(x)Y,dz]'Adzf,

with

J

2n dz'^ix, t) dz" = X —~—^^—dx^, dx'' 7= 1

then (Ot is a real C°° (1, l)-form on M^, and is represented as ^t = i I

gjapizj, t) dzf A dzf

on each Uj x t, where the gj^pizj, t) are C^ functions of {Zj, t). We define a Hermitian metric on M^ by I

g,„^-(z)dz"®dz-^=

I

gj^p{z,,t)dzj®dzf.

{IAD

Similarly we define a Hermitian metric on the fibre of Bt by Z

aj,^izj,tU}[^,

(IAS)

where the aj^p^izj, t) are C^ functions of (Zj, t). Now we apply the results of § 3.5(c) to our case. For any local {p, ^) -form ij/ on Mt, we define *fj/^ by (3.119) with gta^{z) instead of g«^(z). Representing (pt, if/t e if^'^(B,) as in (7.46) on each UjXta M„ we have I aj^fiizj, t)cp^izj, t) A *,iAf (z,-, 0 A.M

= S a^xfiizk, t)(pi(zj,, t) A *,(/?{?(Zfc, 0 A,/u,

on UjXtn UkXt7^0. From now on we omit the index j for simplicity, and write ZA,^ ^AA(^. 0 0<;PJ (^> 0 A ^t(Pj'{Zj, t). We define the inner product of cpt and if/f by (3.139):

(
I a;,^(z,0
350

7. Theorem of Stability

For tfj, e if^'^(J5,), we define ha^t by (3.140) as

{Kt^r{z,t)=-i a''{z,t)^,dJi a^,(z,o*.ruo), T=l

\At = l

/

where {a^^{z, 0)A,;X=I,...,^ = («AA(^5 0)A,1.=I,...,I.- For simplicity we omit the subscript a in t>at and write it as b^. b^ is the adjoint operator of 5^. Namely, we have

Since gja^izj, t) are C°° functions of (Zj, t), the linear operator *f /5 C°° differentiable with respect to te^. Also since Uj^fiizj, t) are C°°, b^ 15 C°° differentiahle. By (3.143), we define • f = 5ft), + b,a,. n is a formally self-adjoint strongly elliptic linear partial differential operator on ^^'^(Bt) with respect to the ifiner product ( , ),. Since dt and b, are C°° differentiable with respect to re A, so is Dr also. Thus { n j r e A} forms a differentiable family. (p e ^^'"^(Bt) is called a harmonic form with coefficients in Bt if bf

= 0}.

Then by Theorem 3.19, we have the following orthogonal decomposition: ^^'^(J5,) = H^'^(B,)en^^'^(B,).

(7.49)

We denote by H^ the orthogonal projection of^^'^{Bt) onto H^'^(JS,). Let G, be the Green operator of D,. Then by (7.8) we have n,G,(p = G,n,^ = (^ - Ht
(p e i^^'^(B,).

(7.50)

(P = Ht(p + UtGt(p = Ht(p + dtt>tGt(p+btdtGt(p.

(7.51)

Consequently any (pe^^'^(Bt)

is written as

Clearly we have H^dtcp = dtHt

Htht(p = t^tHtip = 0.

(7.52)

351

§7.2. bifterentiable Family of Compact Complex Manifolds

We claim that Gt comrnutes with df Proof. For any «/f Gi^^'^"^^(B,), il/ = H,ilf + G,n tip by (7.50). Put (A = dtGt(p, Then since D ,G,

tt>t = 0 implies that t>tGt(p = Gt\>t
b , a = G,b,.

(7.53)

Let A be an eigenvalue of D „ and e an eigenvector belonging to A. Then A||e||? = ( n , e , e ) , = ||a,e||?+||br^||?^0, hence A^O. Thus we can arrange all the eigenvalues A^'^(0 of D^ on ^P'^iBt) as follows:

o^Ar(o^- • •^Ar(o^- • •,

Ar(o->+^.

Now we apply the results of the preceding section to the family {D, | ^ € A}. First we obtain from Theorem 7.2, Theorem 7.7. A^'^(0 is continuous in ^G A. Next we obtain from Theorem 7.3, Theorem 7.8. dimH^'^(Bf) is an upper-semicontinuous function

ofteA.

Let Cl^{Bt) be the sheaf of germs of holomorphicp-forms with coefficients in Bt. Then by (3.148), H^(M,n^(fi,))^H^'^(B,). Hence, Corollary, dim H^(M,, 11^(5^)) is an upper-semicontinuous function of te A. From Theorem 7.4 we obtain Theorem 7.9. / / dimH^'^(B,) does not depend on re A, then Hf is C°° differentiable with respect to teA. Finally from Theorem 7.6 we obtain Theorem 7.10. / / dimH^'^(B,) does not depend on r e A, then G, is C°° differentiable with respect to teA.

352

7. Theorem of Stability

(b) Proofs of Theorems First we shall prove Theorem 4.4 (p. 200). Proof of Theorem 4.4. Since the sheaf ©, of germs of holomorphic vector fields over M^ is just the sheaf €{ T{Mt)) of germs of holomorphic sections of T{Mt), and {T(M^) | r G A} forms a differentiable family, we see from the Corollary to Theorem 7.8 that dim H^{Mt,€{T{Mt))) is uppersemicontinuous in te ^. I Next we proceed to the proof of Theorem 4.1. Put t = (t^,..., t^) and consider the infinitesimal deformation dMt/dt^. Since 6^ = ^ ( r ( M , ) ) , we have H\M,&,)

= H''\T(M,)).

(7.54)

Let (pteH^'^{T{Mt)) be the harmonic vector (0, l)-form corresponding to dMt/dtm by the above isomorphism. Since z" =f%.{^k, t) on UjXtn 17^X19^ 0 , putting Ojkit)= Z — ^ a =l

—^,

ar^

Zfc=A,.(z,-, 0,

dZj

we see that dMjdt^ is the cohomology class of the 1-cocycle {Ojj^(t)}. The corresponding harmonic form (pt is obtained from {Ojk{t)} as follows: Let { p j be a partition of unity subordinate to a finite open covering {Ui} of X Multiplying the equality Ojkit) =

etM-Oijit)

by Pi, and summing up with respect to /, we get Ojk(t) =

lPiencit)-lpAjit). I

i

Put 1,(0 = Z, M : ; ( 0 , and ^^(0 = 1 , pAkit). Then ^,-.(0 = 4 ( 0 - ^ / 0 . Since 6ij(t) = -0ji(t), we have I

i

a =l

(7.55)

Otrn

OZj

§7.2. Differentiable Family of Compact Complex Manifolds

353

Consequently if we write ^j(t) as

a=\

OZj

we have i

ot^

Since z" = z"(x, t) are C°° functions of {x, r), these ^/(z/(x, t), t) are C°° functions of (x, 0 , hence ^"{Zj, t) are C°° functions of (zj, t). Since ^^fc(0 is a holomorphic vector field on UjXtn Uj^xt<^ Mf,dtOjkit) = 0. Therefore by (7.55), dt^M^dt^M on UjXtnUkXt, Consequently there is a vector (0, l)-form if/te L^'^{T(Mt)) on M^ such that on each UjXt^ M„ ^. = a.^,(0.

(7.56)

Clearly a^i/^^ = 0. If we represent (/^^ on each UjXt SLS

>P. = l l 4>Uz, t) dzf^, a (i

4>U^, 0 = ^ ^ % ^ ,

OZj

dZj

then the coefficients if/fpizj, t) are C°° functions of (Zj, t). Thus il/f is C°° differentiable with respect to ^ e A. For simplicity we write Tt for T{Mt) below. Let sd{Tt) be the sheaf of germs of C°° sections of Tt. Then since St = 0{Tt), we obtain from (3.106), H\Mt,et)^nMtJtSdiTt))/dtT(Mt,sd(Tt)),

(7.57)

which is induced from the exact cohomology sequence ' "^nMt,s^(Tt))-^nMt,dtS^iTt))^^

H\Mt,et)

-^0.

Since by (7.56) and (7.55), eAr = 5,{^,(0}, and 8{^j{t)} = {Oj^it)} for the 0-cochain {^j{t)}, we see from the definition of 5* that the cohomology class dMt/dtm of {Ojkit)} corresponds to i/^^Gr(M„a,^(r,)) by the isomorphism (7.57). Since TiMt,~dMTt))

=

n''\T(Mt))®dt^''%T{Mt)),

and

we have nM„dMT,))/d,nM„M{T,))

= H'''\TiM,)),

(7.58)

354

7. Theorem of Stability

whQTQ il/ter{Mt,dtS^(Tt)) corresponds to H,ilj,eH^'\T(Mt)) via this isomorphism. Since the isomorphism (7.54) is the composite of those in (7.57) and (7.58), letting cpt be the harmonic (0, \)-form corresponding to the infinitesimal deformation dMJdtm, we see that (ft = Hfil/t.

Since by (7.51) ij/t = Htif/t-^dtt>tGtil/t-^'i)tdtGtil/t, and since dtCfil/t = Gtdtipt^O, we have ^t = (Pt-^dtVt,


Vt = '^tOfil/f

(7.59)

Now we return to the proof of Theorem 4.1. Under the assumption that dMt/dtm = 0 identically, we must prove that if dim H^{Mt, St) is independent oft e A, we can find a 0-cochain {dj{t)} which satisfies 8{dj(t)} = {Ojk(t)} with each Oj(t)=Y, Ofi^j, t){d/dzf) being a holomorphic vector field on UjXt<=^ Mf such that df{Zj, t) are C°^ functions of {Zj, t). Proof Since dMJdt^ =0,(pt = 0, hence ^t = ^tVt and rjt = t>tGtil/t. Since ipt = dt^j(t) by (7.56), putting 0j{t) =

^j(t)-vt,

we obtain 'dfitit) = 0. Thus dj{t) is a holomorphic vector field on UjXtc: Mt, By (7.55) we have Ok(t)-ej{t)

= ^,,{t)-^j(t)

= ej^it),

that is, 8{ej{t)} = {Sjkit)}. Since dim H^'H T(M,)) = dim H ^ M , 0,) does not depend on rG A, Gt is C°° differentiable with respect to te^hy Theorem 7.10. Since ^t and b, are C°° differentiable with respect to re A, so is 7)t = ^tGt^t, hence, if we write 17^ on UjXt as "

d

Vt= H

«=i

vf(^pt)T-^, dZj

the coefficients rjfi^p 0 ^^^ G^ functions of (Zj, t). Since ^f{Zj, t) are C°° functions of (z^, 0 , we see that

e^(zj,t) = ^f(zj,t)-vn^j^t) are also C^ functions of (Zj, t).

I

Theorem 7.11. Assume that dim//^(M^, B^) is independent of t = {t,,...,tm)eA. If idMt/dt,)t=o^H\Mo,So), K = 1 , . . . , m, are linearly independent, then dMJdt^ e H^{Mt, B^), K: = 1 , . . . , m, are linearly independent for \t\< e provided that e>0 is sufficiently small

§7.2. Differentiable Family of Compact Complex Manifolds

355

Proof. For each K: = 1 , . . . , m, we construct as above the harmonic (0, l)-form (p^, = H,ilj^,en^'\T{M,)) corresponding to dMJdT^eH\M,,(dt) via the isomorphism //^(M^, 6j) = H^'^(r(M,)). Since by the assumption dim H°''( r ( M , ) ) = dim H\M,, %,) is independent of f G A, H, is C°° differentiable with respect to t by Theorem 7.9. Since ijj^t is C°° differentiable with respect to t, so is (p^t = i^t^Kt- If idMt/dt^)t=oe H\MO, SQ), K = 1 , . . . , m, are Unearly independent, the corresponding (p^QeYf'^{T{Mo)), K = 1 , . . . , m, are also linearly independent. Consequently, since each (p^t is C°° differentiable with respect to t, (p^^GH^'^(r(MJ), K = l , . . . , m , are also linearly independent for | ^ | < e provided that e > 0 is sufficiently small, which in turn implies that the corresponding SMJdt^ e H^{Mt, 0^), K = 1 , . . . , m, are linearly independent. I As was seen from Example (iv) of §6.2(b). Theorem 7.11 is not necessarily true if dim H^{Mt, St) varies with t. In the following we state the outline of the proof of Lemma 4.1. First consider H^{Mt, €{Bt)), where {M^ | r G A} is a differentiable family of compact complex manifolds and { B J r e A} is a differentiable family of holomorphic vector bundles over M^ Lemma 7.8. If dim H^{Mt, €{Bt)) = d is independent of teA, then we can choose a basis {0 is sufficiently small. Proof Let {(pu..., ^d} be a basis of H\MO, 0(BO)). We can choose C°° sections il/fq e ^^'^(Bf) of Bt with g = 1 , . . . , J such that il/tq are C°° differentiable with respect to t and that if/oq = (pq. This follows immediately from the fact that if we denote by X the underlying differentiable manifold of B^, then Bt as a vector bundle over X is the restriction of a vector bundle ^ over XxMo X = Xxt, and that ^ is C°° equivalent to BQxA. By (7.49),

where H'''(B,) = {cpe^''^\Bt)\~dt

q=

l,...,d.

Since dim H^'°(Bf) = d is independent of t, H^ is C°° differentiable with respect to t. Since ifjtq are C^ differentiable with respect to t, cptq are also C°° differentiable with respect to t for q = l,..., d. Since cpoq = (pq are linearly independent, (ptqE H^{Mt,€{Bt)) are linearly independent for | ^ | < e provided that e > 0 is sufficiently small, hence {cpti,..., (ptd} forms a basis of

H\M,€{Bt)).

I

356

7. Theorem of Stability

Now consider a complex analytic family { M J r G A} of compact complex manifolds where AcC"" is a polydisk with centre 0. We assume that dim H^{Mt, 0^) = J is independent of r G A. Then by the above Lemma 7.8, we can choose a basis {cpn, • • •, 0 is sufficiently small. Namely, if we represent cptq on UjXt as " d
OZj

then the coefficients
UjXtn

Uj,Xt7^0, ct /

^\

v^ dfjk{^k'> ^ =1

0

ft

and since f]k(^k, 0 are holomorphic in zl,..., d(p^qj{Zj, t) _

ar^

"

^/ffc(Zfe, 0

a^i

/

^x

dZfc

zJJ, ^ j , . . . , ^^, we have a(pgfc(Zfe,

az^

0

ar"^

Therefore a

"

d
d

are also holomorphic vector fields on M^. Hence we have

T^
l^^^J,

(7.60)

where the coefficients a^pq(t) are C°° functions of ^ on |^| < e, Let Cp(t),p = l,..., d, be C°° functions of r on |^|<e, and put 6t = Zp=i Cp{t)(ptp. Then dteH^iMt^St) is C°^ differentiable with respect to ^ Thus, if we write 6t as

a=l

OZj

on UjXtcz Mt, the coefficients ^"(z^, 0 are C°° functions of (z^, t). If ^"(z,, 0 are holomorphic in z ] , . . , zj, ^ j , . . . , r^, we say that dt is holomorphic with respect to t. Since OJ{Zj, t) are holomorphic in z ] , . . . , z", ^^ is holomorphic

§7.2. Differentiable Family of Compact Complex Manifolds

357

with respect to t if and only if dOt/dt^ = 0 for K = 1 , . . . , m. Since by (7.60) d .

ydCpjt)

dt^

p

d

dt^

q

p = l \ot^

dt^

q=i

/

^tp,

^t = Y,p Cp(t)(Ptp is holomorphic with respect to t if and only if d

^

— c^(0+ I a,pq{t)cq{t) = 0, Ot^

p = l,...,d,

K = l,...,m.

q=l

Introducing the matrices A^(t) = (a^pq{t))p^q=i^__^^d, we can write this condition as

' • ' -AM

K = l,,,.,m.

(7.61)

\cAt)i

\Cd{t)l

We must prove that taking a sufficiently small e' with 0 < 8'
^tq=

Z Cpq(t)(ptp,

q =

l,...,d.

p=l

Since dim H^(M„ 6,) = d, if det C{t) ¥^ 0 with C ( 0 = (Cpg(0)p,^=i,...,ci, then {^n,...,^rd} forms a basis of f/^(M„ 6,). Moreover, by (7.61), if C ( 0 satisfies the system of partial differential equations -^C{i)^AMC{t)

=%

K = 1 , . . . , m,

(7.62)

each Qtq is holomorphic with respect to t. The system of equations (7.62) has a solution C{t) with d e t C ( 0 ^ 0 for | ^ | < e ' provided that e ' > 0 is sufficiently small if the integrability conditions ^ A , ( 0 - - | - A . ( r ) + A.(OA,(0-A,(OA.(0 = 0 ot^

(7.63)

dtx

are satisfied for K, A = 1 , . . . , (i (see [21], §19). It follows immediately from the definition (7.60) that the matrices A^t), K = 1 , . . . , m, satisfy (7.63).

358

7. Theorem of Stability

Consequently there exists a solution C{t) of (7.62) with det C(t) 9^ 0. Thus, we can find a basis {On, ^.., Otd} of //^(M„ €),) siich that each Otq is holomorphic with respect to t. I (c) Canonical Biasis and Its Applications In this section we let {Mt\te^} be a differentiable family of compact complex manifolds and {Bt\teA} SL differentiable family of holomorphic vector bundles B^ over M^. Fix a non-negative integer p once for all and consider the linear operator D^ ^S^b^ + b^af acting on the linear space

where n = dim M^. Put

where we defitle A? = D" = 0. Since 5? = 0, for any (p,il/e ^^'%Bt), we have (a,b,<;p, htdtil^)t - ((5?bf<;p, dtil/)t = 0, hence Af and D? are mutually orthogonal. Therefore by (7.49) we have ^P'^B,) = W'%B,)®A'f®Dl

(7.64)

Moreover we have d,Dr'

= Af,

b,A? = Dr\

In fact, since 5,H^'^~'(5i) = 0, and dtAt' A? cz d,^P^^-\B,)

q = h...,n,

(7.65)

= 0, we obtain

= d,Dr' ^ d,h,^'^'(B,)

= Al

hence 5fD?~^ = A^ Therefore

b,A?=t,d,r>r'=b,dt^^^''\B,)=or'. The linear nlap dt'. D ^ ^ ^ A? i^ bijective. For, since any <^ G D?"^ can be written as (p =b,(/^ with ijje^^''^{Bt), we obtain lkll? = (
§7.2. Differentiable Family of Compact Complex Manifolds

359

Similarly we have 0 ^ 0 ? = D?. n , A ? = A?,

n , D ? = Df.

(7.66)

Let \^h\t) and e'lh be the eigenvalues and eigenfunetions of D , on if^'^(B,) respectively such that {e?i,..., e^h,...} forms an orthonormal basis of i^^-'^CB,) with respect to ( . )„ Dtcl = \i^\t)el, and that 0^\[''\t)^"*^Xi'\t)^'",

ki^\t)-^+oo (/i-^oo).

By (7.66) we may assume that each eigenfunctioh e% belongs to one of W'^iBt), A?, and D?. Let {e^J with /ij < • • • < 4 < * • * be the set of all the eigenfunetions belonging to A? among {e?h}, and put M?fc = e?f,fc, «i^^(0 = Al,f (r). Then we have n,ul = ai'\t)ul,

(7.67)

and {w?i,..., w?fc,...} forms an orthonormal basis of A^ with respect to { , )f Clearly we have 0
ai'\t)^-^oo (/c->oo).

Since a,MSc = 0, by (7.67) we have drb,u^k = ai'\t)ul,

(7.68)

hence we obtain

Replacing ^ by ^ + 1 in this equality, we obtain (b,ur,b,M?r). = 5,,ar^>(0. Therefore putting

^'^=V:F^''"'^^'

^'-''^

we have {vlk,vl) = 8uH. From (7.65) obviously we have uffceD?. Since by (7.68)

(7.70)

360

7. Theorem of Stability

we have

Thus v^el^j is an eigenfunction of D^ belonging to a^i^^^\t). Wu ^a, • •'} forms an orthonormal basis ofD^. In order to see this, since Wk,v1h)t = 8kh by (7.70), it suffices to show that any eigenfunction e% belonging to D? is written as a linear combination of f^^'s. Since e%e

Hence we have dtel = l^'^ aj,ut\

a,eC,

k

where the summation is taken over all k such that ai^'''\t) ^r^?h = 0, we have

= x\?\t).

Since

k

Consequently, since by (7.69),

e% is written as ,q -y(h) ^ ,,q e%=r'UkV\,

Ck-

^fc

It is clear from (7.68) and (7.69) that

Put dq = dim H^'^(B,). Then {^n,..., ^?dj forms an orthonormal basis of H^'^(BJ where in general dq depends on t. Since ^^'^(B,)=H^'^(jB,)eA?eD?, {e?i,..., e%^, ufi,..., v^i,...} forms an orthonormal basis of =^^'^(5^). Note that for ^ = 0, we omit w^, w?2, • • •, and that for q = n,we omit Vn, ^"2, • • • •

§7.2. Differentiable Family of Compact Complex Manifolds

361

Theorem 7.12. For each g = 0 , 1 , . . . , n, we can choose an orthonormal basis {e1„...,e\,u''n,...,v'>a,...}

oi ^"'"{B.)

(7.71)

satisfying the following conditions. (i) (ii)

nte% = Oforh = l,...,dq where d^ = dim H^'^(B,); For each q = l,,.. ,n, there is a sequence {a^k\t)} with 0
•,

ai^\t)-^+oo

(/c-^oc),

such that

(iii)

D,ul=ai'\t)u^k, The equalities h,u%=^Jai'\t)

n,v^k =

• vV,

ai'^'\t)v^k;

~dtvl = s/ai'^'\t)'

ut'

hold. Note that in (7.71) we omit u^tk and f"fc. The union of the bases (7.71) for q = 0,... ,n forms an orthonormal basis of ® ^=o ^^'"^{Bt), which we call the canonical basis. By (3.148) we have H^(M„a^(B,)) = H^'^(B,). Put /i^'^(r) = dim H^(M„ n^iB,)) = dim H^'^(5,). Fix an arbitrary 8 > 0, and let N ^ ( 0 be the number of X^h\t) smaller than s. Thus, if h^N\t), then A i ^ \ r ) < e , and if N''(t)
^ = 0 , 1 , . . . , n,

(7.72)

/c = l , 2 , . . . .

(7.73)

where we put p^{t) = ^"""^0 = 0. For any ^o ^ A, choose e such that 0<s
q = h...,n,

By Theorem 7.7, each \\^\t) = Xf^'"^(t) is a continuous function of ^ G A , while by (7.73), \i'\to)> s if Xi'\to)9^0. Thus Xi'\t)<8 if xi^\to) = 0, and Xi^\t) > s if xi^\to) T^ 0 on | r - ^ol < ^ provided that 5 > 0 is sufficiently small. Hence N^{t) = N''{to)

on

\t-to\<8.

362

7. Theorem of Stability

Therefore, since by (7.73) v^ito) = 0 for g = 1, 2 , . . . , n, we obtain from (7.72) hP'''{t)-hp%t)+p''^\t)

= h''''(to)

on \t-to\<8.

(7.74)

Theorem 7.13. If d i m / / ^ " ^ M , , 11^(B,)) and dim//^""'(M,, n^(B,)) are independent of te^, then dim H^{Mt, Cl^(Bt)) is also independent of teA. Proof For any ^o^ ^, if we take a sufficiently small 5 > 0, by (7.74) v'^it)^ v^'^\t) = 0 on\t-to\<8 since by the assumption hP'^~\t) = hP'^~\to) and hP^^^\t) = hP'^^\to). Therefore, again by (7.74), /i^'^(0 = /z^'^(^o) on |r - ^ol < 8. Thus h^'^^it) is constant on a neighbourhood of tgeA. Since to is an arbitrary point of A, dim/f^(M„ 0^(5,)) =/i^'^(0 is independent of te A. I Applying this theorem to 0 , = (^(r(M^)), we obtain the following immediately. Corollary. If dim H'^'^iMt, 6^) and dim H'^'^^{Mt, ©^) are independent of r G A, ^/len dim //^(M,, © J f^ afeo independent of teL,

Appendix

Elliptic Partial Differential Operators on a Manifold by Daisuke Fujiwara

§1. Distributions on a Torus A local coordinate system on an n-dimensional differentiable manifold X gives a diffeomorphism of its domain Xj onto the unit ball U inU", while the unit ball in R" is diffeomorphic to a domain V in the torus T" = [R"/27rZ". Thus a section of a vector bundle B with /i-dimensional fibres over Xj can be identified with a vector-valued function with values in C^ over the domain V in T". Various function spaces of sections of B over Xj can thus be considered as those of corresponding C^-valued functions over V in T". In what follows, we shall always treat the functions defined on V which can be extended to the whole space T". Since the torus T" is compact, a function space consisting of vector-valued functions on T" has much simpler structure than that on R". (a) Definition of Distributions By A: = ( x \ . . . , x") we denote a point in R". If we denote by ITTZ" the set of the points in R" whose components are ITT times integers, then this becomes an additive group. The n-dimensional torus T" is defined as T" = R727rZ". A C°° function/ on T" can be regarded as a C°° function on R", hence we can write it as f{x) = / ( x \ . . . , x"). Then we have the multi-periodicity / ( x + 27r^)=/(x),

^eZ\

(1.1)

We write DjJ = l,... ,n, for the differential operator d/dx\ An n-tuple a = ( « ! , . . . , a„) of non-negative integers will be called a multi-index, and |a| = aj 4- • • • + a„ its length. Moreover we introduce the following notation: D " = D^^ • • • D ^ .

364

Appendix. Elliptic Partial Differential Operators on a Manifold

We denote by C°°(T") the totality of complex-valued C°° functions on T". For / = 0 , 1 , . . . , the norm | |/ is defined by \ip\i= I

max|D>(x)|.

(1.2)

With the topology defined by these countably many norms, C°°(T") becomes a Frechet space, which we denote by ^ ( T " ) . Definition 1.1. A continuous linear map of ^ ( T " ) to C, namely a continuous linear functional on ^ ( T " ) , is called a distribution on T". We denote the totality of distributions on T" by ®'(T"). In other words, S is a distribution on T" if and only if it satisfies the following two conditions: (1) (2)

S\ C°°(T") 3ip^ S{(p) G C is linear with respect to
(1.3)

holds. The totality ^'(J'') of distributions on T" is the dual space of ^ ( T " ) . We write (5, (p) for S{(p) if 5 G ^'("IT") and cp e ^ ( T " ) . < , ) gives a bilinear map ^ ' ( T " ) x ^ ( T " ) ^ C . As the dual space of ^ ( T " ) , we can introduce a topology into ^ ' ( T " ) . Recall that a sequence {Sm}Z=i of distributions converges to S with respect to the weak topology of ^'(T") if and only if {S,
(1.4)

m-^oo

holds for any (pe2D{J"). Let f(x) be an integrable function on T", and put if,
f{x)
(1.5)

for any (p e ^ ( T " ) , where dx = (I/ITT) dx^ - - - Jx". Then { , ) is linear with respect to (p. By putting / = 0, and C = ljn \f(x)\ dx, (1.3) holds for (f, (p), hence by (1.5) / defines a distribution. Thus we have a map

LHT")^^'(ir"). This map is injective. Indeed, l e t / a n d g be elements of L^(T"), and suppose

§1. Distributions on a Torus

365

for any cp e C°°(T"). Then we have

I

(f{x)-gix))cpix)dx

= 0.

Hence we have the equaHty f{x) = gix) = 0 for almost every x. Therefore f=g in V(J"). Consequently the map of L^(T") to ^ ' ( T " ) defined above is injective. By this injection L^(T") can be considered as a subspace of 2)'{J"): L'(T")c: ^ ' ( T " ) . Let a{x) be a C°° function on T", and f{x) an integrable function on T". Then the product {af)(x) = a{x)f(x) also becomes an integrable function. We want to define a multiplication of any distribution 5 by a by extending this operation. In a special case S ' = / G L ^ ( T " ) , we have the identity (af, (p) =

a(x)f(x)(p(x)

dx =

JT"

f(x)a{x)(p{x)

dx = {f, acp)

JT"

holds for any (pe2)(J"), follows:

In view of this identity we shall define aS as

Definition 1.2. Let 5 G ^ ' ( T " ) and a e C°°(T"). ThenthQproductaSe is defined as follows: {aS,(p) = {S,a(p) for any

2\J'')

(;PG^(T").

To make this definition possible, it should be proved that the linear map <|)^:^(T")3(p^a(;pG^(T")

is continuous with respect to the topology of ^ ( T " ) . For any multi-index a = ( « ! , . . . , « „ ) , we have the following identity (Leibniz' formula): D-(a(x)
I

ll)D^a(x)D-^
(1.6)

where ^ = (/3i,..., i8„) is a multi-index, and by j8 ^ a we mean that Pj ^ a^ for every j = 1 , . . . , n. Further we have put

(

;

)

=

(

;

;

)

•

(

;

:

)

•

366

Appendix. Elliptic Partial Differential Operators on a Manifold

Using Leibniz' formula, we obtain for / = 0 , 1 , . . . , \a
(1.7)

where a C°°(T") and (p C°°(T"). Therefore O^ is a continuous linear map. Since a is a continuous linear map, we can define its dual

which is continuous with respect to the weak topology. 6y the definition of the product of distributions given above, we have *^a{S) = aS. Hence the map ^\J'') 3 (p -^ aS e ^'{J'') is weakly continuous. Its linearity in S is also obvious. L e t / 6 C ^ T " ) . Then we obtain the following identity.

This is just the formula of integration by parts, but this characterizes the partial derivative {d/dx^)f of/ as a distribution. By extending this formula, we define the derivative (d/dXj)S of any distribution 5 as follows. First we note that the map - ^ : 2(Jn

3
oX

J = 1 , . . . , n,

(L8)

oX

is linear and continuous. For, in fact,

U-(p ^Wh^u \oX

/ = 0,1,...,

(L9)

1/

holds. Definition 1.3. We denote the dual map of the linear map (1.8) by ^:S'(T")9S^^5eS'(T"),

j = l,...,n,

and call (d/dx^)S the partial derivative of 5 in x^. The map S^(d/dx-^)S continuous in the weak topology of ^ ' ( T " ) .

is

The above definition of the partial derivative d/dx-^ coincides with that of the usual partial derivative of a C^ function / This follows from the

§1. Distributions on a Torus

367

above formula of the integration by parts. So we always write ^

^

•

=

^

as an operator acting on distributions. Leibniz' formula (1.6) can be extended for any aeC°°(T") and

D^aS)=

I

(-^)D^aD"-^5,

Se

(1.10)

P^a \p/

which is called Leibniz' formula as well. Similarly we can define the translation of a distribution 5 in the direction of Xj by h. For (p e C°°(T"), we put r;'cp(x) =
+

K...,xn.

Clearly the map thus defined T7'':^(T")->^(T")

is continuous and linear. As its dual map, TJ':^'(T")95->TJ'5G^Xir")

(1.11)

is defined. We call TJS the translation of 5 in the direction of x' by h. The difference quotient operator Aj* is defined by ^^S = h-\T^S-S) for any distribution S. Especially in case S=fe

(1.12) L^(T"), we have

AJ'/(x) = / i - H / ( x \ . . . , x ^ + / i , . . . , x " ) - / ( x S . . . , x " ) ) . The formula D^AJ* = AJ'Dfe holds, and for any a e ^ ( T " ) and any S e ^'(IT"), we have AJ'(a5) = A>T7''5 + aAJ'5.

(1.13)

Let 5 be a distribution on T", and K a compact subset of T". S is said to vanish on i ^ ' = r " - X if (5,
368

Appendix. Elliptic Partial Differential Operators on a Manifold

the support of S coincides with the support of / as a function: supp 5 = s u p p / ForaGC°°(T"),wehave supp aS = supp a n supp 5.

(1-14)

(b) Vector-Valued Distributions Let ^6 be a positive integer. A vector-valued distribution S with values in C^ is defined as a /x-tuple 5 = (S\ . . . , S'^) G ^'(T") X . . . X ^ ' ( T " ) . We denote the space consisting of all such vector-valued distributions by ^'(T", C^). Since 2'{J", C^) = 2)'{J") x • • • x ^ ' ( T " ) , it is a vector space. For aGC°°(T") and 5 G ^ ' ( T " , C ^ ) , we define their product by aS = {aS\... ,aS^). Similarly we define the partial derivative DjS and the difference quotient AjS by

DjS =

(DjS\...,DjSn,

A ; 5 = (AJ'5\...,AJ'5'^).

We denote the space of vector-valued C°° functions by ^ ( T " , C^) = ^ ( T " ) X . . . X ^ ( T " ) . For (p = {(p\...,(pn^^(T\C^) define the bilinear map

and 5 = ( 5 \ . . . , 5 ^ ) G ^ ' ( T " , C ^ ) , we

A

which makes 2\J",

C ) and ^(T", C") dual to each other.

(c) Fourier Series Expansion of Vector-Valued Distributions We denote by L^(T", C^) the totality of vector-valued functions f{x) = (/^(^), • •. ,f^M) with values in C^ defined on T" such that each component/'^(x) is square-integrable. For / gG L^(T", C^), we define their inner product by a«)=I

A

fix)g\x)dx. JT"

(1.15)

§1. Distributions on a Torus

369

With this inner product L^(T", C^) becomes a Hilbert space. The norm of / is given by

11/11 = «/)''' = ( i f {fixWdx) . For any f G Z", we write i' x = ^^x^ + • • • + ^„x", and put f{x)txp{-i^'x)dx.

f^=\

(1.16)

JT"

We call /^ = if],... , / f ) the Fourier coefficient of / Using these / ^ , we obtain for each A a Fourier series expansion f{x)=lf,

exp(i^- x),

A = 1 , . . . , M,

(1.17)

which converges in L^(T''). We denote by P(Z", C^) the totality of ^c-tuple of infinite sequences of complex numbers {a^} with ^ e Z" and A = 1 , . . . , /i, such that A

^

An element of /^(Z", C^) can be considered as a map of Z" to C^ which maps ^ e Z " to a^ = ( a | , . . . , a f ) e C ^ /^(Z",C^) is also a Hilbert space. The following is the fundamental theorem for Fourier series. Theorem 1.1. The map which associates/e L^(T", C^) to its Fourier coefficients gives an isomorphism o/L^(T", C^) onto /^(Z", C^) as Hilbert spaces. \\fr = ll\fl\' A

(Parseval).

I

(1.18)

^

The Fourier series (1.17) converges to / with respect to the norm of L'(T",C^). It is difficult to give the condition under which a square-integrable function on T" becomes continuous in terms of the Fourier coefficients. But for our present purpose the following theorem is sufficient. Theorem 1.2 (Sobolev's Imbedding Theorem). Let f^ be the Fourier coefficients offe L^(T", C^), and suppose that for some s> n/2, we have

A

^

370

Appendix. Elliptic Partial Differential Operators on a Manifold

where we put \^\^ = ^?+ • • • + 1 ^ . Then the Fourier series

\

.

(1.19)

fix)={f\x),...,r(x)),

converge absolutely and uniformly on T", hence give continuous functions of X. Further

f{x)=f'ix) holds almost everywhere. Moreover there exists a positive constant C^ depending on s such that m a x l / ^ W I ^ c Y z Z d + l^rri/^r)

.

(1.20)

Proof. Putting

•=(ii{i^ey\n?)"\ we have

E i/^Ni(i+i^pn/^i(i+i^iv 1/2

^ii/L(i(i+i^rr)

<00

because s> n/2. Therefore the infinite series (1.19) whose terms are continuous functions converge absolutely and uniformly in x, hence represent continuous functions of x. On the other hand, by the preceding theorem, / and / coincide as elements of L^(T", C^). Consequently on T" the equality

f\x)=f'{x) holds except possibly for a set of measure 0, which proves the theorem.

I

Theorem 1.3 (Sobolev's Imbedding Theorem). Let f) be the Fourier coefficients O / / G L^(T", C^), and k a non-negative integer. If for some s> n/2-\-k,

ll{l-^mVW<^

(1-21)

A I

holds, then the Fourier series / ' ( x ) = 1 / ^ exp(if • x), A

A = 1 , . . . , /x,

(1.22)

§1. Distributions on a Torus

371

and the infinite series obtained by term-wise differentiating them j-times for any j^k all converge absolutely and uniformly. Therefore f^{x) are C \ Moreover on T", the equality

fix)=f\x) holds except possibly for a set of measure 0. Proof. We have already proved that (1.22) converge absolutely and uniformly in the preceding theorem. By differentiating their general term, we have \D"f, exp(i^- x)| = \{i$r-ff

exp(i^- x)|

g|^n/^|, 1

A = 1,...,M,

where we put \a\-j^k, and ^" = ^?^ • * • f^". Therefore, similarly as in the proof of Theorem 1.2, we obtain

i\^nf,mfi(i{i+m'-'Y'Since s>n/2-\-k,

we have S^ (1 + |||^)^"'
A = l,...,)Lt,

converge absolutely and uniformly, and the j{x) positive constant C« independent of/ such that

are C^. Also there is a

max|D«/^(x)|^Cj|/||,.

(1.23)

X

The fact that f{x) =f(x) holds almost everywhere is proved similarly as in the proof of the preceding theorem. I The vector space of all C^ functions defined on T" with values in C^ becomes a Banach space if we define the norm o f / = ( / \ ,f^)^ C^(T",C^)by \J]jc = l

I

max|DTW|.

Let / ^ be the Fourier coefficients of / Then for any multi-index a with I a I ^ /c, we have

(-i)'"'r/^ = [ f{x)D"

expi-ii • X) dx

= - I ( - D ) T ( x ) exp(-»|- x) dx.

372

Appendix. Elliptic Partial Differential Operators on a Manifold

Therefore, recalling {-D)ye

L^J",^),

we obtain

ii:a+\i\'r\m'=m+Di+---+Dim^c,\A,.

(1.24)

By the deep gap between (1.24) and Sobolev's imbedding theorem, we cannot get the precise characterization of C^(T", C^) in this way. But we can characterize C°°(T", C^) as follows. Theorem 1.4. fe L^(T", C^) coincides with an element o/C°°(T", C^) almost everywhere if and only if its Fourier coefficients satisfy

llil

+ my\fe\'<^

(1-25)

for any seU, The proof runs as follows. First by (1.24) we see the necessity of (1.25), Conversely, if (1.25) holds for any 5, we see from Sobolev's imbedding theorem that / is equal to an element of C°^(T", C^) almost everywhere. I Definition 1.4. Let cp^ be the Fourier coefficients of (p = {(j)^,... C°°(T", C^). For an arbitrary 5 e R, we call

,(p^)e

ikiis=(i;z(i+i^r)>^r)''' the Sobolev norm of degree s. By (1.23) and (1.24), for /c = 0 , 1 , . . . , there exists a positive constant Q depending only on k such that for any (p e C°°(T", C^), Ck^\\(p\\2k^\(p\2k^Ck\\(p\\2kHn/2-]+U

(1-26)

where [n/2] denotes the integral part of n/2. Therefore the topology of the Frechet space ^ ( T " , C^) can also be defined by the countable system of norms || H^, /c = 0,1, From this fact combined with Theorem 1.4 and the proof of Sobolev's imbedding theorem, we obtain the following theorem. Theorem 1.5. Let cp e ^ ( T " , C^). The Fourier series of cp converges to (p with respect to the topology ofQ){'f, C^). Therefore {exp(i|^ • x)}^^^" forms a basis of 2(J\C^). I Consider the Fourier series expansion of a vector-valued distribution S = (S\ .,.,S'')e S)\J\ C^) with values in C ^ Definition 1.5. Put 5^ = {S\ exp(-/^• x)), ^G Z", A = 1 , . . . , M^. We call S^ = ( 5 | , . . . , 5^) the Fourier coefficients of the vector-valued distribution 5 =

§1. Distributions on a Torus

373

( S \ . . . , 5 ^ ) . The series X S, exp(-f^- x) = (l S\cxp{-i^'

x ) , . . . , Z 5^ exp(-/^- x))

is called the Fourier series of S. Theorem 1.6. (r) (2°)

The map ^'(T", C ) BS-^iS""^) is injective. 77ie sequence (5^) w the Fourier coefficient of a vector-valued distribution if and only if there exists an integer k^O such that II(l +l^m5^r<^ A

•

(1.27)

^

holds. This being the case, the Fourier series ^S'^Qxpi-i^'x) converges weakly in ^'(T", C"). Let S e 2)'(J", C^) be its limit. Then the Fourier coefficients of S coincide with S^. Proof ( r ) It suffices to show that if 5 = ( 5 \ . . . , 5^) e 2)'(T", C^) and for any^eZ^ 5^ = {S\ exp(-i^ • x)) = 0,

A = 1 , . . . , /x,

holds, then 5^ = 0 . But this is clear since {exp(-/^- x)} forms a basis of (2°) By the definition of 2)'(J", C^), there exist a positive integer / and a positive constant C such that for any f eZ", K 5 \ exp(-/^- x))| ^ C1exp(-/^- x)b holds. Hence by taking a suitable constant C,

holds. Thus if we choose k so that / c - 1 > n, (1.27) holds. Conversely, suppose that (1.27) holds for some k^O. If we put A^ = (1 + If p)~'=Sf, then we have

II|A^P
374

Appendix. Elliptic Partial Differential Operators on a Manifold

Hence if we put A^{x) = Y.^^^ exp(~i^- x), then this becomes a squareintegrable function of x Thereifore A^{x) represents ^ distribution on T", namely, for any cp e ^ ( T " ) , {A\
A\x)
Y.A^^Qxp{-ii'x)(p(x)dx JT"

^

holds. Since thg series I ^ A^ exp(-if • x) converges strongly in L^(T"), we can exchange the integration witlj X^, so we have {A\cp) = lA'^ ^ ^

exp(-f^-x)<^(x)^x = IA^_^, JT" JT"

^

where we denote the Fourier coefficients of (p{x) by (p^. Next, we put a partial sum of the series made of {5^} as S'N=

Z

5^exp(-/^-x),

where N>0. This is a C°° function, hence gives a distribution on T". We shall prove that if N ^ o o , 5 ^ converges to the distribution (1 - D j - • • • Dl)^A^ with respect to the weak topology of ^'(T")- In fact, if (^ G ^ ( T " ) , we have <(1-D?

Dl)'A\
I

= {A\il-D',

exp(/^-x)(l-D?

DD'cp) D^JVWrf:^

JT"

Applying (1.25) to <^(x) € C°^{J", C ) by putting 5 = fc, we have

ikf.=i(i+i^m.p-,r<^. Therefore we have m-Dl

Dl)''A\v)-{S%,cp}\

= \ I 5,V-^ \|^|>iV

/

\|^|>iV

/

§1. Distributions on a Torus

375

hence by (1.27) we obtain lim(5^^,(p) = < ( l - D ?

Dl)^A\).

N-»oo

Thus SN converges weakly to ( 1 - D ? DD'^A'' in ©'(T"). If we put 5^ = (1 - Di DD'^A^, then the Fourier coefficients of S" are given by

{S\cxpi-ii-x))

= {A\{l-D]

Di)''exp{-i^-x)}

=ii+\enA\cxp{-i^-x)) Thus the proof is completed.

|

Theorem 1,7. For any S = {S\ . . . , 5^) e ^'(T", C^), there exist a non-negative constant / ^ 0 qndf= (f\ . . . ,f^) e L^(T", C^) such that in the sense of distributions S' = {l-Dl

DlYf

(1.28)

holds. (The structure theorem of distributions.) Proof, Let 5"=I5^exp(/f-x). be the Fourier expansion of S^. Then (1.27) holds for some k^O. Therefore as in the proof of the preceding theorem, we put

/^=(i+i^r)-'s^ If we p u t / ^ ( x ) = X^/^ exp(f|- x), then/^(x)G L^(T"), and as is shown in the proof of Theprem 1.6, we have

S' = (l-Dl

dl)T

in the sense of distributions. By putting / = /c, the theorem is proved.

I

Theorem 1.8. Let 5^ be the Fourier coefficients ofSeSD'iJ", C ) , and (^^ the Fourier coefficients of cp e S)(J", C^). Then {S,
(1.29)

I

This formula is already obtained in the proof of the preceding theorem.

Appendix. Elliptic Partial Differential Operators on a Manifold

376

(d) Sobolev Space We introduce a space which Hes between ^ ( T " , C^) and ^'(T", C ) . Definition 1.6. For any 5 G IR, we put

and call it the vector-valued Sobolev space of degree s with values in G^, where S^ denotes the Fourier coefficient of 5^. For S,Te

W^(T", C'^), we define their inner product by (S,T),=llil + \^\ysi¥,,

(1.30)

which makes \y^(T", C^) a Hilbert space. Here T^ denotes the Fourier coefficient of T^. The norm of this space is just the Sobolev norm of degree s 1/2

i5L=(ii(i+i^rri5^r) Theorem 1.9. W'{J\C^)cz

(V)

Ifs'<s,

(3°)

Ifk-\-n/2<s,

W^'(T",C^).

there exists a continuous imbedding W'{J\

C^)-> C\J\

C^),

(1.31)

namely, there exists a positive constant C^k such that for any S e C^) with W^(T", £^), there is an SeC\J\ |5U^C,,||5||,.

(1.32)

(4°) ^ ( T " , C ^ ) is dense in each \^'(T",C'^). Proof (1°) is clear from the definition. (2°) follows from Theorem 1.4 and Theorem 1.6. (3°) is just Sobolev's imbedding theorem, Thorem 1.3. Finally, if we put \¥'(T", C^), and let S = (S\ ...,S'')e 5"=i:5^exp(/^-x) be its Fourier series, then the partial sum X|^|siv ^e exp(i^- x) is an element of ^(J\ C^) fnd converges to S in W'(T", C^) for N^oo. (4°) follows immediately from this. I

§1. Distributions on a Torus

377

The following inequality is often used in the following. Lemma 1.1. Let a, b, t be positive numbers. If

O^X^l,

a^b^-^^Ar^/^a + ( l - A ) r ' / ( ' - ^ ^ 6

(1.33)

holds, where the equality holds for t^^^a = t~^^^^~^^b. The proof is given by the comparison of the arithmetic and geometric means. I Proposition 1.1. (r)

Ifs"<s' < s, then for anyfe inequality holds:

W'(J", C^) the following interpolation

ui'^mi'-'"'^''-'"^^^^^^^ (2°)

(1.34)

Ifs"<s'<s,foranyfe T^'(T",C^) and any positive number t, the following interpolation inequality holds:

s-s" s-s'

+

^-u-.vu-.;||y||...

(135)

s-s"

Proof By the above lemma, for any ^ > 0, we have

ii+\^\Y^^^^t^'-'"^^^''-'"\i-h\^\y

P u t / = ( / \ . . . , / ^ ) , and let/^ be the Fourier coefficients of F^. Multiplying the above inequality by |/^|^ and summing over ^ e Z " , we obtain

Taking the minimum of the right-hand side for r > 0, we obtain

Taking the square root, we obtain (1.34). (2°) follows from this and Lemma 1.1.

I

378

Appendix. Elliptic Partial Differential Operators on a Manifold

Theorem 1.10. Ifs'<s,

the inclusion M^'(T",C^)^ l^^'CT^C^) is compact.

Proof. It suffices to show that if a countable sequence {fm}m in W'(J", C^) is bounded, we can choose a subsequence which converges in W (¥", C^). Let fm = (fL --^Jm) and frn,^ the Fourier coefficients of fl where ^eZ\ Since {/m}m is bounded in \y^(T", C^), there exists a positive number M such that

ll/ll?=iiiA/(i+iirr<M. Since for each ^, A, {/^, Jm=i is a bounded sequence, we may take a suitable subsequence {m'} so that for each ^, and A, {/^,m'}m' converges. Then {f^'} converges in \^^ (T", C^). In fact, we have the following Fourier expansion: /'^' = Z/m',,exp(/^-x). For any positive number e, take a sufficiently large N so that

holds. Then for any m', we have the following inequality.

Fix such N. Since there are only a finite number of lattice points ^ e Z " with 1^1 ^ N , we may choose a sufficiently large mo so that for any mj, m^>mo, A|^|<7V

holds. Then we have

11/.,-/mill?'^I z

\fU,,-fUf(^+\^\y'

A i^i<;v

+ 2E I (IA./ + l/k/)(l + l^lY A l^l^iV

Therefore {/mlm' is a Cauchy sequence in W"^(T", C^), hence converges in W''(J\ C^). This completes the proof. I Since the Sobolev space of degree 0 is nothing but L^(T", C^), we shall write II II for || ||o below. It is difficult to have an intuitive understanding

§1. Distributions on a Torus

379

of \y^(T", C^), but if 5 = / is a positive integer, it is rather easy to understand. For this we use the following lemma. Lemma 1.2. Let S = {S\ . . . , 5^) G £ ) ' ( T " , C ^ ) , and 5^ the Fourier coefficients ofS. If we write the Fourier coefficients ofDjS = {DjS\ ..., DjS^) as (D,5^)^ forj = 1, 2 , . . . , n, we have iDjS'), Proof,

= i^jS'^,

j = l,...,n;

(DjS^)^ = {DjS\ exp(-/^- x)) = -{S\ = i^j{S\exp{-i^'x))=i^jS',.

A = 1 , . . . , M-

(1.36)

Dj exp(-/f • x)) I

From this lemma we obtain the following proposition. Proposition 1.2. Let I be a positive integer. Then W\J\

C^) = {Se L^(T", C^)ID'^Se L^(T", C^) for any a with \a\^ /}.

For any Se W\J",

C^) we have 1/2

n-'''\\S\uJ

|iiD«5ii; |D"5||)"'^||5||,.

I

(1.37)

Mais

Proof It suffices to show (1.37). Letting 5^ be the Fourier coefficient of 5, we have

For

^GZ", \a\^l

holds. Therefore, multiplying this by |5^|^ and summing over ^, we obtain n-^\\S\\l^

I

\\D-Sf^n^\\S\\l

\a\^l

which proves (L37). Theorem 1.11. (1°)

The restriction of the bilinear form { , ) on ^\J\ C^) x ^ ( T " , C^) to ^ ( T " , C ^ ) x ^ ( T " , C^) gives the following inequality: For any (t>,il/e ^ ( T " , C ) and any seU, K(A,^)|^||eA||-slkL.

(1.38)

380

(2°)

(3°)

Appendix. Elliptic Partial Differential Operators on a Manifold

The bilinear form ^ ( T " , C^) x ^ ( T " , C^) 3ip,(p^ (lA,
sup^^||S||_.

(1.39)

Here sup is taken over all non-zero cp eQ)(J'',C^). Proof (1°) Let cp^ and il/^ be the Fourier coefficients of (p and ip respectively. Then we have A

^

By Schwartz' inequaUty, we obtain

^ll'All-.lkL. (2°) Since ^(T",C^) is dense both in W-'(J",C^) and in W'{J\C^), the inequaHty (1.38) shows that the bilinear form ( , ) extends uniquely to a continuous bilinear form on W"'(T", C^) x W'(J\ C^) to C. Then (1.39) proves that by this bilinear form, \ y ( T " , C^) and \y'(T", C^) become dual to each other. (3°) Let S^ be the Fourier coefficients of 5. For any N>0, we define
Then \{S,
\\
||
\ A l^l^iv

=1 I \s'A-^+m-'. 2\-s|

c'\\2\l/2

I|«lsiv (l + l^r)"'!'^^!)

/

Taking the supremum with respect to N, we obtain (1.39).

I

> we have

§1. Distributions on a Torus

381

Proposition 1.3. Let a be a multi-index. If s> s' + \a\, then for any Se W'{r,C^), D'^SeWXl^C'^). D " : \ ^ ^ ( T " , C ^ ) ^ \ ^ ' ' ( T " , C ' ' ) is continuous, and the inequality ||D"5i|,,^||5||,^,,«l

(1.40)

holds. Proof. It suffices to show (1.40). If we let 5^ be the Fourier coefficients of 5, the Fourier coefficients of D"5'^ are given by

{D''s)',={iirs',. Hence we have

iio"5ii^.gii(i+iin^iii^i"i|s^rgiisii?.,i„|. I Next we consider the product of a function and a distribution. Lemma 1.3. Let f(x) e C°°(J",C''^) be a C^ function with values in {px fi)-matrices, and put fix) = if\{x),...,fj,{x)). Let g = (g\ ..., g^)e W\J'',C^) be a vector-valued distribution with values in C^. Suppose that

p= l

holds in the sense of the product of distributions. Then h = {h^,... ,h'')e \y^(T", C^). Moreover there exists a positive constant C depending only on s such that the following inequality holds:

ii/iii,^cm,,,iigii,. Proof The above theorem is true for any seU, but its proof is complicated. Therefore we shall give proof only for the case s is an integer here. This is enough for our present purpose. (1°) The case 5 = 0. The assertion follows from the well-known inequality

JT"

) g ' ' ( x ) | ^ ^ x s m a x | / ^ ( x ) | | |g''(x)|Vx ^ JT

with C = l. The case 5 ^ 1 . For a multi-index a with | a | ^ 5 , we have by Leibniz' formula P^a

\p/

382

Appendix. Elliptic Partial Differential Operators on a Manifold

Hence letting D^f be the (^ X/i)-matrix function with D^/p as its (A, p)component, and D'^'^g the C^-valued distribution with the pth component D'^'^g^, and applying the result of the case 5 = 0 to D ^ / and D'^'^g instead o f / and g respectively, we obtain D«ft||g I

(f\\D'f]o\\D''-''g\\

by taking the summation with respect to j8. Therefore, by Proposition 1.2, there exists a positive constant Ci such that

(T) The case 5 = - / < 0 . Take an arbitrary «p7^0 with (p^ \^^(T",C"). Using (1.39), we have II/i II _f = s u p - — p - = sup-7—r-,

I

In particular in case 5 ^ 0 , ^'W=Z/pW^'W,

A = l,...,^,

(1.41)

p

hold for almost every x, Theorem 1.12. Let I be a positive integer with l^n-\-\. Then for any fe W\J\C^^)^ and geW'iJ^C''), /i = (/i\ . . . , / i ^ ) given by (1.41) is an element 0/ V^^(T", C ) . Moreover there exists a positive constant C depending only on /, n, /i, v such that the following estimate holds: \\hh^c\\fUg\\i-

§1. Distributions oh a Toms

383

Proof. Using the estimate of ||i^"/i|| for the inulti-indice^ d with |a| ^ /, we can find an estimate of ||/i||/. By Leibniz' formiila we have

|^|^|a|/2 \ P /

l^l^kl/2 \ P /

Since | a | ^ / , if | j 8 | ^ | a | / 2 , then |j8| + (n + l ) / 2 ^ / , hence, by Sobolev*s imbedding theorem, there exists a positive constant C independent o f / such that on T" |D^/^(x)|SC||/||, holds outside of a set of measure 0. Therefore we have

||ID^/^D"-VNC||/|M|D"-^g||,

|)8|S|a|/2.

P

Similarly in case |j8| > | a | / 2 , since |a - / 3 | < |a|/2, applying Sobolev's imbedding theorem to g^, w6 obtain

IIID^/^D-VNCIID^/Il ||g||„

|)8|>|a|/2.

Therefore

||D"ft|| s c(||/|M|g|||„|+||/|||„|||g||,) g c||/|M|g||, holds. Thus by Proposition 1.2, we obtain

|i/.||,sc||/|M|g||,. I We define a linear partial differential operator A(x, D) which operates on vector-valued distributions S = {S\ .,.,S'')e ^'(T", C^) by (A(x,D)Sr=l

I p

a^a(x)D«5^

A = l,...,^,

(1.42)

|a|gm

where the coefficients Up^ix) are C°° functions. If there are some a with |a| = m. A, p, and a point x such that apa(x) T^ 0, we say that A(x, D) is a partial differential operator of order m.lfm = 0, A(x, D) is called a multiplication operator. For a non-negative integer /, we put Mi= I

I Zmax|D^a^«(x)|.

(1.43)

Propositidn 1.4. Ler A(x, D) be a linear partial differential operator of order m with apa(x) being C°° functions. Then for any seU, there exists a positive number C depending only on s, n, m, v and fjL such that for any S e W'{J",C^),

Appendix. Elliptic Partial Differential Operators on a Manifold

384

the following inequality holds: \\A(x,D)Sl^CM\4Sl^mThis follows from Lemma 1.3 and Proposition 1.3.

I

Let (p(x) e C°°(T", C), and (p the operator of multiplication by (p(x): For 5^)-^ {(pS\ . . . , (pS^). We will often use the 5 6 ^'(T", C^), (p:S = (S\..., commutator of (p and A(x, D) later: [A(x, D), (p] = A(x, D)

(1.44)

In terms of components, this is written as ([A(x,D),(p])^=S

Z

a^„(x)[D",(p]5^A = l , . . . , / x .

(1.45)

Lemma 1.4. (1°)

For a multi-index a, ^/le commutator [D", (p] i5 a //near partial differential operator of order \OL\ — \ whose coefficients are derivatives of(p of order up to\a\. For any seR, there exists a positive constant C depending on s and a such that for any Se iy^(T", C^), the following inequality holds: ||[D'',.p]S|l,SC|.p||,H„ll|S||,,l„l_i.

(2°)

For such a linear partial differential operator A(x, D) as stated in Proposition 1.4, the commutator [A(x, D),
\\[A(x, D), cp]Sl^

CMH|(^|,,,^^||5L^^_I

holds. Proof (1°) We proceed by induction on \a\. First let |a:| = 1. Then i[Dj,
J = l,...,n,

hence, there is a positive constant C such that \\[Dj,
§1. Distributions on a Torus

385

a multi-index a of length / - 1 , and 7 = 1 , . . . , n, we have

hence, by the hypothesis of induction we obtain ||[D,D«, -\S\U,^\m,

^]D«5||,

^C|J|,4-/||*S'||, + /_i, which proves (1°). {T) follows immediately from (1°).

I

Next we examine the relation between the difference quotients and the Sobolev spaces. The results obtained will be used in §6. Theorem 1.13. (r)

For S e W'(J", C^), the difference quotient Af 5 is again an element

(2°)

For any h with \h\ ^ 1, ifSe

l ^ - ^ T " , C^), we have

||AtS||,^||5||,^i, (3°)

IfSe

W'(J\C^),

j=h...,n.

and for any h with 0 9^\h\^l

IIAJ'SII.^M holds, then Se W'^\r,C^)

(1.46) and j = 1,...

,n,

(1.47)

and

||5||Li^«M+||5||?

(1.48)

holds. Proof. (1°) Let S'=lS'^cxp{i^-x) be the Fourier expansion of 5. Translating S in the direction of Xj by -h, we obtain r7^5^ = 1 5 ^ Qxp(i^jh) ' exp(/^- x), hence the Fourier coefficients of rJ^S are given by S^ Qxp(i^jh). Since |Sj| = |5^exp(/^,/i)|, we have | | T 7 ^ 5 | | , = ||S||,. On the other hand, the

386

Appendix. Elliptic Partial Differential Operators on a Manifold

difference quotient is given by A^S=h-\r;'S-S\ hence for /i 7^ 0, we have A^Se ^^(T", C^). (2°)

^^S' =1 S',h-\txp{ih^j)-l)

hence, using the inequahty \Qxp(ih^j)-l\^\h^j\,

exp(/^. x), we obtain

siii5^r(i+iirr' A e

S\\S\\U^. (3°)

Aj"5^ = Z 5^p(ft, e ) exp(i|^- x),

where we put pih,^j) =

h-\expiiHj)-l).

By the hypothesis (1.47), we have

A

i

By Fatou's lemma,

Msiiminfiii5^np(/i,f,)r(i+krr g I I l i m i n f | 5 ^ r | p ( / , , $ ) r ( l + |^|Y

A

^

Therefore we obtain

A e

^ 11511, +wM^

I

Proposition 1.5. Set 0<s'
§1. Distributions on a Torus

387

eachj=l,...,n, I

{A^SfsH'^''

dh< 00

(1.49)

holds. This being the case, there exists a positive constant C = C(s\ 8) depending on s' and 8 such that the following inequality holds:

c-'iisii?,,,siisiiHi

/•oo

j J-oo

\\A^sr\hr'''dh^c\\s\u..

Proof. Since Hrj'SlI^ = \\S\\s, for h> 8,WQ have |A;5||?^2;I-^||5||,.

Hence

[

\\Afsry-'''dh^2\\srs\

J\h\>8

h-'-'-uh=^\\s\\i

J\h\>8

S

Therefore the condition (1.49) is equivalent to the following inequality. /•oo J —oo

Let 5^ be the Fourier coefficients of S. Then we have

MS r

ll\S',\'\cxp{ihij)-l\\l

+

\ey\hr''-'dh.

J-oo A ^

By Fubini's theorem, changing the order of the integration and XA Ii^» we obtain

where we put a{$j)-

\cxp{iMj)--mhr'-'dh.

Thus a{$j)>0, and for f eR, we have ait^j) =

\cxp{iht^j)- l|^|/i|-^^-' dh = I f ^ ai^j).

Therefore there is a positive constant C = €{$') such that

388

Appendix. Elliptic Partial Differential Operators on a Manifold

Consequently there is another positive constant C = C{s') such that C(5')-'(l + | | | T s i + a ( | . ) + - - - + a ( ^ J g C ( 5 ' ) ( l + |f|Y', C(s'nS\\U,.^ll\S',\^l

+ \eni

+ a{^,)+--

• + a(^„))

= ||si|,+i f" \\^^sfy-''dh j

J —oo

^C{s')\\S\\Us'.

This completes the proof.

I

The following proposition and Lemma 1.5 will be used in §2. Proposition 1.6. Let z be a point of T", and B^iz) the ball of radius r with centrez: B^z) = {xeT"|X,- (xj-Zjf^r}. Supposes^ 1. Forcpe W^(T", C^) with supp (pc: Br{z), we put ,(z,r) = sup-7——.
\\(p\\s

Then N^iz, r) does not depend on z. If we put Ns(z, r) = N^ir), we have limN,(r) = 0.

(1.50)

Proof. It is clear that N^iz, r) does not depend on z. Suppose that (1.50) is false. Then there is eo > 0 such that for any positive integer /, there exists
\\(pi\\s-i^eo'

Since {cpi} is a bounded sequence in W'{J", C^), by taking a subsequence if necessary, we may assume that {(pi} converges to cp in W*~^(T", C^). Since 5 - 1 ^ 0, we have cp e L^(T", C^), but supp cpi a B^/iiz) = {z}. Therefore ^ = 0 as an element of L^(T", C^). On the other hand, since || (Pi\\s-i = ^o in W^"^(T", C^), we have Hc^H^.i = lim/_^oo||

Let / ^ O be an integer. Then there is a positive function R{r.,l) such

§1. Distributions on a Torus

389

that for any cp G \^'(T", C^) with supp (p ^

B,(z),

\\acp\\i^R{rJ)\\
(ii)

Rir, I) ^ r\a\^ + Ar|z|(r)|a||„,

/ ^ 0,

hold. Moreover we have lim^^o ^ ( ^ /) = 0. Ifl = —kisa negative integer, there exists a positive-valued function C{r, I) such that for any (p e W\J'', C^) with supp (p a B,(z), ||a(p||/^2r|a|i||(p||f+C(r,/)|a|2|/|||^||/-i holds.

Proof (i) Note that for

XG JB,(Z),

\a{x)\^r\a\, holds. First suppose / ^ l . Then since / is a positive integer, there is a positive constant C = C{1) such that WacpW^^n'^' Y. \\D-a
I

( | | a D > f + ||[a, D«](pf)

^C(r|a|i||(p||, + |ah||^||;_i). Using Proposition 1.6, we obtain \\acpl^C{r\aU

+ Ni(r)\ah)\\
hence we have

R{r,l)^C(r\a\,-hNM\ah). In case/ = 0, ||(p||/^ r|a|i||
| | S | | i _ . = ||(l-A)-'^'^S||L

390

Appendix. Elliptic Partial Differential Operators on a Manifold

Take a function fe C°°(T") so that 0 ^ / ( x ) ^ 1, / ( x ) = 1 on B^z) ^nd that f{x) = 0 outide of B2r(z). Then / ^ = (p. From this and (*), we have \\ap\U = \\il-Ar'^'aM\=

A+ B,

where A = ||a/(l-A)-''^^(p||, and B = \\[af, {1 -A)-''^^](p\\. Since s u p p / c B2r(z), using (*) we obtain AS2r|aM|(l-A)-'=/VN2r|aU|,.||_,. On the other hand, since [af, (1 -A)-*^^^] = {l-Ar''\af,

(1 -A)''/^](l-A)-'^/^

we obtain

where we use Lemma 1.4. By (*) and Theorem 1.12, we have B^C(r)\a\2,\\
C(r) = C\f\2,.

Combining the estimate of A, we complete the proof. If / = -fc is odd, we have only to use ||5p_,-||(l-A)-^'^^^^/'5f + i

\\DJ{l-^)-^^^'^^^Sf

7=1

instead of (*).

I

Finally we prove the following proposition. Proposition 1.7. Let K be a compact set of T", and let {L(,}/=i a finite open covering of K. Take a)j{x)e C°°(T", C) with suppaj>yc: Uj so that on some open set G containing K, J j

Then for any s, there exists a positive constant C such that for any Se W'{J", cn with supp 5 c: X, C-'\\Sl^i

\\cojSl^C\\Sl 7=1

holds. C = C(s) may depend on the choice of G and co.

§2. Elliptic Partial Differential Operators on a Torus

391

Proof. Since there is a constant C with || co^S || ^ ^ C || 51| „ the second inequality is obvious. The first inequaUty follows from the existence of positive numbers C and C such that

||5||, = ||n(x)-^n(x)5||,^c||a5||,^c'i||ci;^5||,. i §2. Elliptic Partial Ditfereiitial Operators on a Torus (A) Estimates by the Sobolev Norm (a) Elliptic Operators with Constant Coefficients We set (A(D),(p)^= Z

Z a^al)>^

A = l,...,/..

(2.1)

|a|^/, p = l

A(D) is called a linear partial differential operator with constant coefficients, operating on vector-valued distributions witk values in C^. When a p „ # 0 for some a with |a| = /, we say that A{D) is of order /. The sum of the terms with \a\ = l

A(D)= I Ia^«D«

(2.2)

\a\ = l p

is called the principal part of A{D). For any ^G Z" and w = (w^ . . . , w^) e C ^ we have 9 = (exp(/^-X))WG D(T", C^). Let w'= {w'\... ,w'^) be defined as w' = exp(-/^- x)A(D){exp(ff • x)w}.

(2.3)

Then we have

^"= I K(/^)"w-. The correspondence w^w' gives a (/x X;u)-matrix valued polynomial A(i^) in ^ of degree /. Let A{i^) = {a{i$Yp). Then w'^-(A(/^)w)^= I

Ia^p«/'"lr<

namely,

A(i^) is called the characteristic polynomial of A{D).

(2.4)

392

Appendix. Elliptic Partial Differential Operators on a Manifold

We obtain from Ai{D) the matrix Ai(i^) of homogeneous polynomials of degree /, whose (p, A)-component is equal to E|at=/^P«('^)"- ^i(^i) is called the principal symbol of A(D). Definition 2.1. When the principal symbol Ai(i^) is a regular (/x x^)-matrix for every s e Z", we say that A{D) is of elliptic type. Besides the supremum of positive constants satisfying

(lk'1')"'s5|^|'(s|w1^y'' is called the constant of ellipticity ofA{D),

(2.5)

where we put w"^ = {Ai{i^)w)^.

Lemma 2.1. Suppose that Ai{D) is an elliptic differential operator of order I which consists of only its principal part, and let SQ be its constant of ellipticity. Then for any seU and any (p e W^'^\J", C^), the following inequality holds: ||A,(D)(^||?+5?||(p||^^2-^5?||
(2.6)

Proof Let (P''=1
exp(i^- x),

p=

l,...,fi,

be the Fourier expansion of cp = (cp^,..., (p^) e 1^^(T", C^). Then, using (2.2), we have {A,(D)cpr=l

I

Ia:«(/^)>?exp(i^.x)

= i' 1 iAiU)
A = 1 , . . . , M.

Hence, by the definition of the constant of ellipticity, we have

\\MD)
e

^ A

Therefore we have |A;(D)^||?+5?||^||?^5?II|<^^r(l + | ^ r r ( l + l ^ n ^ A

:2

Thus we have proved Lemma 2.L

0o||
§2. Elliptic Partial Differential Operators on a Torus

393

In (2.1) put l = 2m. Then by Proposition 1.3 we find that for any (p e W"^(T",C^), A{D)(pe U^""^(T",C^). Therefore by Theorem 1.11 (2°), we can define a Hermitian form on 1^"'(T", C^) by putting {AiD)^,il,)= I

ma',^D-
\a\^2m

(2.7)

A p

for (p,ilje \y"^(T", C^). Especially for cp, if; e C°^iJ\ C^), we have {AiD)cp,ij;) = {A{D)
(2.8)

where ( , ) denotes the inner product of L^(T", C^). Since

(A(D)^,.A) = I 1 laU\

D>-(x).A'W dx,

we can rewrite it by partial integration as

iA{D)
I

IbU, D > ^ ( x ) D ^ ( A W ^ x .

(2.9)

A \a\,\l3\^m p

Here there appear no derivatives of order greater than m in

V ^ ) G C ^ and w ' e C ^ set

^ =. ^^-^-w - (^'^•"w\ . . . , e'^-^w^), where ^€Z". We define a bilinear form as follows: (A(D) e'^-^w, e'^-"wO = (e-'^-M(D) e'^-X >v') = Z(A(/^)w)V\

(2.10)

A

This is the Hermitian form on C^ associated to the matrix of characteristic polynomial of A{D). The sum of the terms of the highest order is equal to (-l)"X(A2.(f)w)V^

(2.11)

X

which is the Hermitian form associated to the matrix of the principal symbol ofA(D). Definition 2.2. If the real part of the Hermitian form (2.11) associated to

394

Appendix. Elliptic Partial Differential Operators on a Manifold

the matrix of the principal symbol of the linear partial differential operator (2.1) is a positive definite form, we say that the differential operator A(D) is of strongly elliptic type. The maximum of such positive numbers 8 that ( - 1 ) - Re Z

(A2.(^)W)^M;^

^ 8^1^^ I \w'\'

(2.12)

is called the constant of strong ellipticity. Lemma 2.2. If a strongly elliptic partial differential operator with constant coefficients AjmiD) of order 2m contains no terms of order ^ 2 m —1, then we get, for (pe 1^^"(T",C^), Rc{A2rniD)
(2.13)

where 8o is the constant of strong ellipticity. Proof For cp = ((p\ ...,(p^)e

^^'"(T", C^), let

(p^=Z(p^exp(i^-x),

A = l,2,...,/x

be the Fourier expansion of (p. Using (2.10), we have Re(A2^(D)(^, ip) = i-ir

Re 1 1 iA{^)
since A{D) consists only of the principal part. Therefore we have R^iA2m(D)
+ \^D\
A e

^i-dlMl This completes the proof. (b) Elliptic Linear Partial Differential Operators with Variable Coefficients Let l[7 be a domain of T". We define a differential operator A{x, D) with C°° coefficients defined on an open neighbourhood of [(7] as follows: For (PGC°°(C/,C^),

(A(x,D)(p)^(x) = Z I

<*' (pax ) D > - ( x ) ,

(2.14)

§2. Elliptic Partial Differential Operators on a Torus

395

where apa(x) is a complex-valued C°° function defined on an open neighbourhood of [ U], and for some index a with |a| = /, ap^M does not vanish identically. / is called the order of A(x, D). The part of the highest order A/(x, D) of A(x, D) is given by {A,{x,D)
I

a^„(x)D>-(x).

(2.15)

We call Ai{x, D) the principal part of A(x, D). Let w = ( w \ . . . , w^) G C^,/(x) a real-valued C°° function defined on (7, and t 3. positive number, and put

Substituting this for (p, we see that e-''^^"U(x, D)(e''-^("V)

(2.16)

is a polynomial of degree / in t. The term of degree / in t comes from the principal part, and its coefficient is given by (A(x,Uwr=l

I

<(x)(f^J«w^

(2.17)

where ^^ = ( ^ i , . . . , ^J) is given by df(x) = e.dx' + '"-hr.dx\

(2.18)

which is a cotangent vector of t/ at x. Namely, A/(x, ^^x) is a (/A X ^)-matrix which is determined by giving a point (pc, df(x)) on the cotangent bundle of C/. This ()Lt X/x)-matrix-valued C°° function A/(x, f^^) defined on the cotangent bundle T* U is called the principal symbol ofA(x, D). If we take ^- X as a function/(x), the value of the principal symbol is equal to (A,(x,ff)w)^=I

Z <(x)(f^)w-

(2.19)

P |a| = /

and this is the part of e - ' " % ( x , D)(e'"-^w)

(2.20)

of order / with respect to ^. Definition 2.3. Suppose a linear partial differential operator A(x, D) of order /, is defined on a neighbourhood of [[7]. We say that A(x, D) is of elliptic type, if there is a positive number 8 such that for any point (x, ^x) T^U

396

Appendix. Elliptic Partial Differential Operators on a Manifold

with ^x 7^ 0 and for w 6 C^, the principal symbol satisfies the inequality l\Mx,i^)wr\'^S'\i^fl\w'\\ A

(2.21)

A

The maximum of such 8 is called the constant of ellipticity. In other words, A(x, D) is of elliptic type if and only if for every point Zoe[U], the differential operator A(zo, D) derived from A(x, D) by replacing the coefficients ap^(x) of A(x, d) by ap«(zo) is of elliptic type in the sense of Definition 2.1. We want to show the corresponding fact to Lemma 2.1 concerning elliptic differential operators with variable coefficients. We introduce the following quantity: for /c = 0 , 1 , 2 , . . . , Mfc=IZ

I

maxjD^a^«(x)|.

(2.22)

Also we use the following notation: Ct{ L/, C^) = {(p G C^(T", C^) I (p(x) = 0 for any x e U}, W'oi U, C^) = {the closure of C^( L/, C^) in W'(J\

C^)}.

Theorem 2.1 (Local Version of the a priori Estimate). Let A{x, D) he an elliptic linear partial differential operator defined on a neighbourhood of [ L^] and let 8Q be its constant of ellipticity. Then for sel. there is a positive constant C = C(s, 6, M,,|) such that for any cp e W'o^\U,C'') ||
(2.23)

Proof In order to prove this theorem, it suffices to show (2.23) for any cp e C^iU, C^). We choose a sufficiently small e such that 2enVMi<2~^-^/^6o. Cover [L'^] with open balls Bi,..., Bj of radius s. Furthermore we choose real-valued C°° functions OJJ(X)J = 1,...,J, defined on T" such that supp (Oj c: Bj and that Y.j (^jM = 1 on [ U]. For any (p e C^( U, C^), we set (pj(x) = (Oj(x)(p(x). Since J

\\
\\
there exists some 7 such that the following inequality holds: \\cpjl^i^J-'\\cpl^,

(2.24)

§2. Elliptic Partial Differential Operators on a Torus

397

Let Pj be the centre of the ball Bj. We write the principal part Ai(x, D) of A(x, D) as in (2.15), and let Ai(pj, D) be the partial differential operator with constant coefficients obtained from Ai(x, D) by replacing its coefficients ^paM by their values a^podPj) at ;?,. This is an elliptic differential operator with 8Q as the constant of ellipticity. Hence (2.6) holds. Therefore we obtain the following: ||A,(p,-,D)(p,-||, + 6oi|(p,L^2-^/Xlk,L+/. .

(2.25)

On the other hand, since {{Ai{x,D)-A,{pj,D))
Z Z«(x)-a^«(/7,))D>;(x),

by Lemma L5 in §1, there exists a positive constant C ( E ) such that \\l{a'pAx)-a';,^{pj))D-
(2.26)

Further, since A{x, D) — Ai{x, D) is a differential operator of order S / - 1 , there is a positive constant Ci depending only on s, I, /x and n such that \\iA,(x, D)-A{x,

D))cpjl^C,M^4
Combining this with (2.26), we have \\A,{pj, D)
D)cpj + {A,{x, D)-A{x,

D))cpj

-{A,{x,D)-A,{pj,D))
+ 2-'-'^'do\\iPj\U,.

(2.27)

Since the commutator [A{x, D), co] is a differential operator of order S / - 1 , there is a positive constant C2 such that \\[A(x, D), w > , . | | , g C2M|,|||.p||,+,_i.

398

Appendix. Elliptic Partial Differential Operators on a Manifold

Hence \\A(x, D)
D)
+ \\[A{X, D),

a>,.]
s||ft>,.A(x,D)(p||, + C2M|,|||
(2.28)

Since there exists a positive constant C3 depending only on s, I, n, fi such that \\o>jAix,D)(plsC3\\Aix,D)
+ C2M|,||i^||,+,_i + nVC(e)M2|,|||(p,||,+,_i + 2-^-'/^5o|k,||.,,. By transposing the last term of the right-hand side, we obtain 3 ' 2-'-^^'8o\\
D)
+

SolWjl

+ CiM|,| II ^^-II ,+^_i + C2M|,| II (p II,+/_! -\-n',jLC{e)M2i4(Pj\\s+i-i,

(2.29)

Since we can take a positive constant C4 such that ||(p^-||,^C4||
||(Py||5_,/-i^C4||
we get by (2.24) and (2.29), 3'2-'-'^'8oJ-'\\
(2.30)

where we put C5= ClC4M|5l^-C2M|^l + nV^(^)^^2|5|• Taking a suitable tin Proposition 1.1, we obtain a positive constant Cg depending only on s, /, n and /JL such that C5||(p||,^,_i^2-^-^/^5o/-^||
I

§2. Elliptic Partial Differential Operators on a Torus

399

Put / = 2m and let w = ( w \ . . . , w'^) and w' = (w'\ . . . , w'^) be vectors in £^. For each (x, ^^) e T* U, we can define a Hermitian form in w, w' (-l)'"I(A2.(x,^Jw)V^ A

associated to the (/x x^)-matrix A2m(^, i^x) for the principal symbol of A{x,D), Definition 2.4. We say that A(x, D) is of strongly elliptic type if the above Hermitian form associated to the principal symbol satisfies the following condition: There is a positive number 8 such that for any (x, f^) G T* (7, ^^^ T^ 0 and for any weC^, ( - 1 ) - R e S (A2^(x, ^Jwrw'

S S^l^.p- I |w^p.

A

(2.31)

A

The supremum of such 8 is called the constant of strong ellipticity. Theorem 2.2 (Local Version of Garding's Inequality). Let U be a domain of T". Suppose that a linear partial differential operator depending on a neighbourhood of [ V] is of strongly elliptic type and that 8 is its constant of strong ellipticity. Then there exist positive constants 8^ and 82 depending only on 8, m, fjL and M^ such that for any (p e W^{ U, C^) Re(Aix,D)
(2.32)

holds. Proof For any (p,ilfe CT{ U, C^), (A(x,D)(p,(A)=

I

I

I

aU^)D-cp'{x)ilfHx)dx.

\a\^2m p,A J T "

By integrating the terms with | a | ^ m + l by parts (|Q:|-m)-times, we can rewrite this into the equality containing no partial derivatives of order ^ m + 1 as follows: (A(x,D)vp,(A)=

I

1 I

b',^^{x)D-
(2.33)

where bp^p{x) is a linear combination of derivatives of order at most m of ^p'a'(x). The sum of the terms with |a| = |/3| = m is given by (A2^(x,D)(p,(A)=

Z

I

I

| a | = |/3| = m p,A J t"

fc^«^(x)D>^(x)DVW^x

(2.34)

400

Appendix. Elliptic Partial Differential Operators on a Manifold

Let A2m{x, ^x) be the matrix of the values of the principal symbol of A(x, D) at (x, f^) E T* t/. For w, w' G C , we have ( - I ) ' " Z (A2.(x, ^Jw)^w'^ =

Z

Z ^:«^(x)^r^>v"w'\

(2.35)

In fact, let / ( x ) be a real-valued C°° function defined on U and g(x) any complex-valued C°° function v^ith support in U, Put ^^^ = df{x) with
(A(x) = e^'^^'^w'

into (2.33), by the definition of A2m(x, ^x) we obtain:

I ^r^'"(A(jc, - 2 m ^ ^ ^ ^ D)g(x) n W ^ v - ^ .>''/(^)w .,''/(^)w' = lim e"^^^'w, e"-'*^'w') t-*oo

l^"

=

|a|Sm.|;8|Sm p,A J T "

I

I

[

fe^.^(x)(i^J«(-i|J''w''w'^g(x)^x.

|a| = |^| = m p,A JT"

Since g(x) is arbitrary, (-l)""I(A2.U^x)w)V^= A

I

I

C^(x)ir^w''w'^

|a| = |^| = mp,A

holds at each x. Since every non-zero element of T* U is written in the form of df{x), we obtain (2.35). For any e > 0, we can cover [ L^] by a finite number of open balls Bi,..., Bj of radius e. Let Pu • - • ^PJ be the centres of Bi,..., Bj, respectively. We may assume that s is so small that the inequality en^'^M^^,<2~'^-^8^

(2.36)

holds. Take real-valued C°° functions a>y(x),7 = 1 , . . . , / on T" with supp (Oj cz Bj such that l(Oj{xy=l,

xe[Ul

(2.37)

For any cpeC^iU^C^), put (pj(x) = (Oj(x)(p(x). Then supp (pja Bj, Now denote by A2m(Pp D) the partial differential operator with constant coefficients obtained from (2.34) by replacing bp^pix) by its value at x = Pj,

§2. Elliptic Partial Differential Operators on a Torus

401

i.e. we put (A^^iPj, D)cp(x)r =

1

1

hU,{Pj)D-^'^'{x).

(2.38)

Then we see by the hypothesis of strong ellipticity and (2.35), that A2miPj, ^ ) is a strong elHptic differential operator with the constant of strong elHpticity 8. By Lemma 2.2, we get Re(A2.(;7„D)(p„
(2.39)

By the way, we have Re(A2„(x, D)(pj, (pj) g Re{A2„{pj, D)
(2.40)

We shall take an estimate of the second term of the right-hand side. ( A 2 „ ( x , D)(pj,
=

I

I

|a| = |/3| = m p,A

[

D)
{bUi^)

- b^„^(p,))D>j'(x)DV;(^) dx.

JT"

Since |x -p^| g g for x supp (pj, we have P

Hence we obtain the following estimate: | ( A 2 ^ ( X , D)ipj,

CPj)-{A2m{Pj,

D)
9,-)N

en'^M^+llk.llm.

By this inequality and by (2.36), (2.39) and (2.40), we obtain Rt{A2rn(x,D)
(2.41)

Hence by (2.37), iA{x,D)cp,
I

II

| a | S m p,X J |0|sm

= l{A{x,D)
+ Z

I Z [ b',^^{x)[wj, D'']
\a\^m p,k j J

+ I

Z Z I b'^^,(x)D-cp}(x)[coj, D^]
\a\^m p,\ j J

Appendix. Elliptic Partial Differential Operators on a Manifold

402

By Lemma 1.4 there is a positive number Cj depending only on m, /JL, n, and 8 such that Re(A(x,D)(p,(p)^IRe(A(x,D)(p, -,(p,-)-CiM^||(p|U||(p|U_i.

(2.42)

j

On the other hand, D)
Re(A(x, D)
D)cpj,
-{A2m(x,D)(Pj,(Pj)\ Making the difference of (2.33) and (2.34), we have D)(pj, (pj)

((A(x, D)(pj, (pj)-{A2mix,

= T

b^«^(x)D>^(x)DVW^:^

l\

\a\^m p,\ J T"

where the summation X' is taken over all the terms either with or with |j8| ^ m - 1 . There exists a positive number C2 such that |(A(X, D)(Pj, (Pj)-{A2m{x,

\a\^m~l

(pj)\^C2Mm\\(Pj\\m\Wj\\m-l-

D)(Pj,

Combining the inequality with (2.41), we can find a positive constant C3 depending only on m, JJL, and 8 such that Rc{A(x, D)

By the above and (2.42), we have

hjWi-S'iyjf

Re(A(x, D)
J

J

\\(Pj\\m\\(pj\\m-l

-C^Mml, j

-C,Mj
(2.43)

There exists a positive constant C4 such that ||(;p^|U^C4||
||(p^-|U_i^C4||
On the other hand Xj ll<^7ir= ll
§2. Elliptic Partial Differential Operators on a Torus

403

Therefore putting C^= C^JCl+ Q , we get by (2.43)

R^(A{x,D)cp,cp)^2—'Cfj-'8^cp\\l-8'\\<pr -QM^||(p|U||(p|U_i.

(2.44)

By (1.34) in Proposition 1.1, we have

By choosing a sufficiently small t in (1.33), we can find a positive number C7 such that

MJ
(2.45)

Hence (2.32) holds for any (peCoiU.C^)Since Co(U,C^) is dense in W^iU.C'^) and both sides of (2.32) are continuous in (peW^iU,C''), I (2.30) holds for every (p e W^{ U, £^).

(B) Estimates by the Holder Norm (a) The Case of Constant Coefficients Set 0 < ^ < 1 . We say that a vector-valued function (p(x) = (
I

V

Z

| D V ( X ) - D V ( X O | _ ^

sup

——,


The totality of such functions (p is denoted by C ^ ^ ^ ( T " , C ) . For (pe C^^\J\ C^) we define the norm by . . ^ V

kk+« = k l f c + Z

V

I

sup

A = l \a\ = k | x - x ' l ^ l

|DV(X)-DV(XO|

,

ne

•

(2.46)

|X —X j

^fc+0^ jn^ C^) is then a Banach space. The same statements as in Lemma 2.1 can be obtained by replacing || \\s-i and II \\s by | Ifc+e and | Ik+i+e, respectively. The proof for the general case is, however, somewhat long. So we restrict ourselves to the most simple case. We assume that the order of the elliptic differential operator is 2 and

404

Appendix. Elliptic Partial Differential Operators on a Manifold

that A2(D) is of diagonal type, namely, we assume that A2iD) satisfies

{A2(D)
i

a^jD.Djcp'ix),

A = l,...,/x.

(2.47)

Furthermore, we suppose that a^ is constant with a^ = aj; for /, j = 1, 2 , . . . , n and that the operator is strongly elliptic. In other words, there is a positive constant 8 such that, for any ^ G R " -Re i

a^^,^j ^d^l

fi.

A = 1 , . . . , At.

(2.48)

By the assumption that A2(D) is of diagonal type, the proof is reduced to the case /x = l. In what follows, putting /it = 1, we shall consider the strongly elliptic partial differential equation A{D) acting on C^'^^^^(T", C) A{D)
(2.49)

Let A = (a'^) be an {n x n)-matrix with a'^ = a^\ Let B = (a^) be its inverse matrix. We introduce the quadratic form on R" by Q(x) = la,jx'x\ u

(2.50)

Define the constant C(n) by C{n)(n-2)

\

Qizy^'^' da,(z) = 1,

(2.51)

J|z!=i

where dai(z) stands for the surface element on the unit sphere \z\ = 1. Set E(jc) = C(n)(?(x)^'-"^/l Then AE(x) = -(n-2)C(n)(?(x)-"/^(la,x^),

(2.52)

and DiDjEix) =

(n-2)C(n)Q(x)-"^^ x(-a.^ + n ( ? ( x ) - ^ ( l a , . , x ^ ) ( l a y ) y

(2.53)

§2. Elliptic Partial Differential Operators on a Torus

405

From these equalities, we can easily see that A{D)E{x)

=0

for X G R" with X9^0. Let p ( 0 be a C°° function if t such that p(r) = 1 for \t\ | . Put ^{x) = p{\x\). If we put g{x) =

ax)Eix),

then we find that g(x) = 0 for |x| ^ | , and that Djg{x) = ax)DjEix)

+ Djax)Eix),

D,Djg{x) - ax)D,DjEix)

+ co,j(x).

(2.54) (2.55)

Here (Oij{x) denotes a C°° function with SUppWyC:{x||^|A:|^|}.

Thus there is a C°° function w such that for x G R" with X9^0, A(D)g(x)

= w(x)

(2.56)

and that SUpp W C l { x | 5 ^ | x | ^ | } .

We extend the functions g(x), cuy(x) and w{x) to the periodic C°° functions on R", and identify them with those on T^ We denote such function on T" by the same notation g{x), Wy(x) and w(x). g{x) is called a parametrix for the differential operator A/(D). In the general case, it is rather difficult to construct a parametrix. Lemma 2.3. Setf{x)

= A(D)u{x)

u(x) = -{

for u e C^(T", C^). Then we have

g{x-y)f(y)d:y-\-

w(x-y)u{y)dy.

(2.57)

JT"

Proof. Set 0 < ^ < 1. Put a , = {>; G T" I ly - x| > s} for a fixed x. By means of Green's formula, u{y)A(Dy)g{x--y)d-y

A(Dy)u{y)g(x-y)dyn,

{Av'Dy)u(y)g{x-y)da{y) \

{Ap-Dy)gix-y)u{y)da{y),

(2.58)

406

Appendix. Elliptic Partial Differential Operators on a Manifold

where Dy stands for the differentiation with respect to y, and (Av Dy) = X a'-'V,((9/^>'') with ^ = ( ^ 1 , . . . , ^„) the outer unit normal vector to dO.e'do'^y) denotes the surface element of afl^, and A{Dy)g{x-y) = w{x-y). Since \{Av' D ) M ( > ^ ) [ ^ C|M|I and \g{x-y)\^CQ(x-y)^^-''^^^ on aHe, we have lim

{Av D)u{y)g{x~y)

dcT{y) = 0.

e^O J .

On the other hanc}, since x-y \\x-yy""\x-y\)'

\x-y\ by (2.52), we get (AvDy)g{x-y) for x,y with 0<\x-y\
=

in-2)C{n)Qix-y)-''^'\x-y\

Jience we have by (2.51) {Av D)g{x-y)u(y)

daiy) = u{x).

Therefore putting e -» 0 in (2.58), we obtain u{x) = - \

g{x-y)f{y)dy-h\

JT"

w{x-y)u{y)

dy.

I

JT"

Lemma 2.4. As for the integral transformation Hf{x) = jj^ h{x-y)f{y) we get the following. (1°)

Ifhis

integrahle and f is continuous, then Hfis continuous and Hf{x)=[

h{y)f{x-y)dy.

Moreover the following estimate holds: \mo^N{h)\f]o, where we put N{h)=\

\h{x)\dx.

dy,

§2. Elliptic Partial Differential Operators on a Torus

(2°)

407

If for j = 1, 2 , . . . , n, Djh is integrable, then Hf{x) is of class C \ Set Ci =\-jn \h{x)\ dx + Y^jljn \Djh{x)\ dx, then we have

l«/l. = C,\f\o. (3°)

Put fcso. If Wh is integrable for any a with \a\Sk C'^iJ", C) and

then Hfe

\Hf],SC\f]o where C =

l^„^^,NiD''h).

The proof is easy, hence it is omitted. By Lemma 2.3 we write M(X) ^ Gfix) + Wu{x),

(2.59)

where G/(x) = - |

g{x-y)fiy)dy

(2.60)

wix-y)u(y)dy.

(2.61)

and WM(A:)= JT"

By Lemma 2.4, for any integer /c ^ 0, we have \Wu\,^CMo

(2.62)

where C , = I | ^ , ^ , N ( P " w ) . Since Djg{x) is integrable for any j , we obtain from Lemma 2.3, DjGf(x) = - \

Djgix-y)f{y)dy=\

^g{x-y)f(y)

L e t / G C ^ T " , C ) . Thqn, by integration by parts, we obtain

^Gf(x) = -l f,( ^ - > ' ) — / ( j ' ) ^ J ' JT"

ay,-

Hence, f o r / e C\J", C), we have, for J,J = 1 , . . . , « ,

D,D,.G/(x) = - [

D,g{x-y)^f{y)dy.

dy.

408

Appendix. Elliptic Partial Differential Operators on a Manifold

Since {dldyj)f(,y) = {dldyj){f{y)-f{x)), D,DjGf{x) = - I

JT"

¥ui QL^={y eJ"\\x-

we obtain

D,g{x-y)

^{fiy)-f{x)) dyj

dy.

y\> e). Then

D^DjGfix) = -lim I D , g ( x - y ) ^ { f { y ) =-o Jn. dyj

-f{x))

dy.

By integration by parts, we get DPjGfix)

= lim i •^D,g{x-y){f{y) «-o Jn^ dyj •limf

By the way for 6 with 0
k'-yl \D,g{x-y)\ \x~y\

-f{x))

dy

i ^ ^ Ag(x->')(/(}')-/W)dcr(g).

(2.63)

there exists a positive number C such that \f{y) -f{x)\

S C\x -

yr-^'^e.

Therefore the second term of the right-hand side of (2.63) vanishes. Consequently, we have

D,DjGf(x) = I ^D,g{x-y)ifiy) =-[

-f{x)) dy

DjD,g(x-yKfiy)-f{x))dy.

(2.64)

The following fact is important for this integral transformation. Lemma 2.5. Suppose that 0 < e ^ i Then DjDig{z)dcr,{z) = 0,

f,j = l , 2 , . . . , n ,

(2.65)

^1^1 = 8

where dcr^iz) denotes the surface element on the sphere \z\ = e. Proof. Suppose 77(0 is a C°° function of teU with v(t)^0 on \t\<2~^ and that 7/(r) = 0 on | r | > i Then we get D,g{z)Djrj{\z\) dz = \im

oo>M = IT"

^^0 J|z|se

such that 7^(0 = 1

D,giz)Djrji\z\)

dz,

§2. Elliptic Partial Differential Operators on a Torus

since

vanishes on |z| <2"^. Besides, since g{z) = E{z) on |z| < 2 " \

D,T/(|Z|)

I

M^=

409

D,g{z)DM\z\)dz

}\z\>e-

DjD,g(z)v{\z\)

dz-

J\z\^e

V{t)t-' dt

-vis)

f-A-g(z)T7(|z|) dcT^z) J\z\ =

DjDiE{z)

e\Z\

da,(z)

I ^D,E{z)dcT,iz). J\z\ = l\z\

Hence M, + T){E)

f-D,E(z)da,(z) V{t)r^ dt

DjDiE(z)

da.iz).

Putting e ^ 0, we see that the left-hand side converges to a finite value. Since

[' Vi](t)t

^ dt = oo^

Jo Jo

we must have DjDiE(z)da,(z)

= 0.

J|z| = l

Thus we see that, for any s with 0 < |e| < | , DjD,giz)dcr,{z)

= 8-'p{8)

J\z\ = e

DjD,E(z)da,(z)

= 0,

J|z| = l

holds, which completes the proof of the lemma.

I

Lemma 2.6. Suppose that 0 < ^ < 1. I f / e C^(T", C), then for ij = 1, 2 , . . . , n DjDiGFe C^(T", C), and there exists a positive number C such that \DjD,GJ]e ^ CN,{g)\J]e,

iJ = 1, 2 , . . . , n,

(2.66)

where we put N,{g)=

I

sup |zr|D"g(z)|+ I

\cc\^2 0 < | z | ^ l

sup |zr^^|D«g(z)|. (2.67)

|a| = 3 0 < | z | ^ l

410

Appendix. Elliptic Partial Differential Operators on a Manifold

Proof. For 1 > p > 0, we set /(x,p)= I

Dp,g{x-y){f{y)-f{x))

dy.

Denoting by o-{n) the area of the unit sphere, we have /(x, p) ^ N,{g)\J]e [

I z r - dz = e-'a(n)N,(g)\f]ep'.

(2.68)

Since g{z) = 0 on |z| ^ 1, for p > 1, we have I{x,p) = Iix,l)

= DjD,Gf{x).

Hence by (2.68) |D,AG/lo^ ^"V(n)N3(g)|/|,. Next suppose \x-x'\ = E<JO. We shall estimate First we write DjD,Gf{x) = [

D,-Ag(x -}^)(/(y) -fix))

D,D,G/(x')= I

(2.69) DjDiGf{x)-DjDiGf{x').

dy,

DjD,g{x'-y)ifiy)-f{x'))

ay +

Iix',5s).

J\y-x'\>5e

By Lemma 2.5, we can rewrite DjDiGf{x') as follows: DjD,Gf(x')=

I

D , A g ( x ' - j ) ( / ( > ; ) - / ( x ) ) tfK + /(x',58)

}\y-x'\>5e

+

DjD,g{x'-y)dy.

{fix)-fix')) \x'-y\>l/2

Hence DjD,Gfix) - DjD^Gfix') = J, + J^ + J,- lix', 5e),

(2.70)

where /, = I

{DjD^gix -y)-

DjD^gix' -y)}ifiy)

-fix))

dy,

J|>'-x'|>5e

J2=\

DjD^ix-y)ifiy)-fix))dy, J\y-x'\<5e l\y-x'\<

J, = Mifix)~fix')),M

=

DjD^giz) dz. |z|>l/2

(2.71)

§2. Elliptic Partial Differential Operators on a Torus

411

If \y-x'\ ^ 5 | x - x ' \ , then the line segment connecting y-x with y-x' does not pass through singular points to g{z). Thus, by the mean value theorem, we see that there is a t with 0
J

{x^-x'')Dj,DjD,g{tx-^{l-t)x'-y).

k=i

\y-x'\>

5s = 5\x-x'\

implies | x - } ; | > 4 | x - x ' | , hence

\tx-^{l-t)x'-y\^\x-y\-\x-x'\^2~^\x-y\. Therefore we have \DjD,g(x-y)-DjD,g(x^-y)\^2"'-'N,(g)8\x-yr-\ Since £ < 10~\ we see that \x-y\^l-^ DjDig{x'-y) 7^0. Hence

|/il^ f

s
for y with

DjDig(x-y)-

2"^'N,{g)8\x-yr'\Ae\x-y\'dy

J\y-x\>4e

S 2"" V(n)N3(g)m« • e(l - 0)-'(4e)«-'

(2.72)

since \f{x)-f(y)\^\f\e\x-yf there. On the other hand, since | > ' - x ' | < 5 e implies that | j ' - x | < 6 e , we have

U,l^

\DjD,g{x-y}\\f{y)-f{x)\dy \y—x\<6e

SN,ig)\f]e\

\zr"dz=e-'N,{g)\f\e(6er.

(2.73)

J|z|<6e

Applying (2.68), (2.72), (2.73) and \J,\S\M\\f[ee' \DjD,Gfix) - DjD,Gf{x')\ S

to (2.70), we get

C2N,(g)\f\ee'

for e < 10~\ In the case of e g 10^', since |D,.D,G/(x) - DjD,Gf{x')\ S 20|D,A.Gyio we finally see that there exists a positive constant C3 such that for |x — x'| ^ 1, |D,D,G/(x) - DjD,Gf{x')\ g CN,(gM,\x

- xf

holds. By this inequality and (2.69), we obtain the lemma. I

412

Appendix. Elliptic Partial Differential Operators on a Manifold

Lemma 2.7. LetA2{D) be a second-order strongly elliptic partial linear differential operator with constant coefficients of diagonal type which operates on vector-valued functions with values in C^. Suppose that its lower terms are equal to zero. Assume that 0<0<1. For any integer /c^O, there exists a positive constant C depending only on 6, k, n, fi, and the coefficients ofA2iD) such that, for any u e C^^^^'{J\ C^),

Proof In case k = 0, the lemma follows from Lemmas 2.3, 2.4 and 2.6. For general /c, since we have A 2 ( D ) D " M ( X ) = L>"A2(D)M(X)

for \a\ = k, by the estimate for the case of /c = 0, we have \D-u\2^e^C{\D-A2(D)u\e^\D-uU). On the other hand, since for any positive integer /c, there is a positive constant Ci such that C^'M^^e^

I

\D"u\e^CMk^e,

\a\^k

the lemma is proved for general k.

I

Remark. In the general elliptic case, we have to construct a (yiix^)matrix-valued function E{x) corresponding to that for the above case such that A{D)E(x) = 0. But is takes a long procedure. The other treatment is quite similar to the above. (b) The Case of Variable Coefficients Lemma 2.8. Suppose that 0 0, the following interpolation inequalities hold:

Proof We have only to prove the case /x = L (1°) The case 0 < r < 5 < L We set A=

sup \x-x'\^l

\x-x'\~''\(p{x)-(p(x')\

§2. Elliptic Partial Differential Operators on a Torus

413

for (p e C^(T", C). There exist points x and x' with \x-x'\ ^ 1 such that \x-x'\-''\(p{x)-(p(x')\>2-'A. Hence we obtain \x - x t S 2A-\\
(2.74)

On the other hand, since

\x-x'r\
hence we obtain A ^ 4^'~'^^'2'^'\(p\'/'\(p\l-'^\ inequality |<^|o^|
Combining this with the trivial

\
(2.75)

(2°) The case 0 < r < s = l.By virtue of the mean value theorem, we have |x - x'\-'\(p(x) - (p{x')\ ^\x-

xf'-'^^'-M,.

By this and (2.74), we see that 2-'A^(4A-'\
(2.76)

(3°) The case r ^ 1 < s < 2. Put 5 = 1 + ^. There exist a point XQ and j such that M =

'ZmdLx\Dj(p{x)\^2n\Dj(p(xo)\. J xeT"

Let y be the point obtained from XQ by the translation in the direction of x^ by r > 0. By the mean value theorem, we find ^ between XQ and y such that cpiy)-(pixo) = tDj
(2.77)

414

Appendix. Elliptic Partial Differential Operators on a Manifold

Since .p 6 C'-'^CT", C), \Dj\Dj
and therefore MS2n(2<~*|^lo+ t^\(p\i+e)- Taking the minimal value of the right-hand side with respect to t, we see that there exists a positive Cg such that

So we can choose a positive constant C'e^l

such that

llr''^

(2.78)

(4°) The case r = l<s = 2. For
(2.79)

By (1°), (2°), (3°) and (4°), we have proved the lemma for r g 1, s g 2 . The general case shall be proved by applying them repeatedly as follows: Suppose 0 < fl < 1. For some positive constants Cj, C2, C3 and C4, we have \Dj
^c,i\DM\i'r'lDMo'''^'Y\Dj
I^K,,sc,kl
Suppose that 0<0
For a,/G C^(T",C),

| a / | , ^ | a U m o + |a|o|/1..

(2.80)

§2. Elliptic Partial Differential Operators on a Torus

(2°)

415

Let A{x, D) be a linear partial differential operator of order I which operates on C^-valued functions. A(x, D) is given as follows: {Aix,D)cpnx)=l

I

aUx)D''
A |a|g/

For 0 with 0<6<1, (2.22) and put Mfc+^ = Mfc+X

and for any integer /c^O, we define Mj^ by

X

I

sup

^

A,p | a | ^ / |/3|^fc | x - x ' | < l

;7^

.

\X — X\

Then for any integer /c ^ 0 there exists a positive constant C depending only on /, /c, 6, n and fx such that | A ( x , D)(p\k+e ^

CMk+e\
The proof is easy. We leave it to the reader. Lemma 2.10. Let z he an arbitrary point of T". Let Br{z) = {x6T"|S^. (x^-z^)^
(2.81)

Proof Since cp is equal to zero outside of B^iz), for any a with \a\ = k and any x e Br(z), we have |D>(x)|^r^|D>|,. Hence we get (2.81), because \a\^k-1

implies \D"(p(x)\^ r\D"(p\i.

I

Lemma 2.11. Let A be an operator of multiplication, i,e., {A
a',{x)
p=i

Suppose 0 < ^ < 1 and let k be a non-negative integer. Suppose that a^s C'^+'CT", C) and that a^(z) = 0,

A,p = l , 2 , . . . , ^ .

Then, there exists a positive constant C depending on k, 0 and fi such that \A
416

Appendix. Elliptic Partial Differential Operators on a Manifold

for all (p e C^'^^(J\ C^) with supp (p c B,(z), where we set K, = S^, ^ \aXfor any s^O. Proof. There exists a positive constant Ci such that

From (2.81) we get |A(p|fc^QrXkk+..

(2.82)

On the other hand, for a multi-index a with |a| = /c, we have

Hence there is a positive constant C2 such that |D"A(PU^I

max | a ^ ( x ) | | D > | e + X,|D>|o+C2i^.+,|(^k_i^,.

A,p l ^ - z N r

On the other hand, since, for |x — z| < r,

holds, we have \D-A | , + Ke\
Applying Lemma 2.10, we see that there exists a positive constant C3 such that \D"A Combining (2.82) with this, we obtain Lemma 2.11.

I

Let L^ be a domain of T". Suppose that a second-order linear partial differential operator A(x, D) with C°° coefficients defined on an open neighbourhood of [L^] operates on
I

aUx)D-cp'(x),

(2.83)

We say that this operator is of diagonal type in the principal part, if ap^(x) = 0 for A 7^ p and \a\ = 2.

§2. Elliptic Partial Differential Operators on a Torus

417

Theorem 2.3. Let U be a domain in J". Suppose that the second-order linear partial differential operator A{x, D) with C^ coefficients defined on [ U] is of diagonal type in the principal part and strongly elliptic. Let 8 be its constant of strong ellipticity on [U\ Suppose 0<6<\. Then for any integer k^O, there exists a positive constant C depending on /i, /x, k, 6, 8 and Mj^+0 such that for any cp e C^-^^^^{J\ C^) with supp (paU, \cp\k+2^e ^ C(\A(x, D)9U+, + |(p|o). Proof This is verified in an almost similar way to Theorem 2.1. We have only to use the norms | 1^+^ and | \k+2+e ii^ place of || ||^ and || \\s+b respectively. We cover [U] with open balls B^, B2,..., Bj of radius s. Let Cj be a positive constant in Lemma 2.11. For any ze[U], let €2(2) be a constant in Lemma 2.7 for the elliptic linear partial differential operator obtained from the principal part of A(x, D) by replacing coefficients by their values at z. Take e so that for any ze{U\ CiC2{z)e^Mk+e<2~^. Choose C°° functions cOj{z), j = 1,2,... ,J, on T" such that supp coj c= Bj and that I^. (Oj(x) = 1 on [ U]. Put
I'Pj

k+2+e

J

We see that for some /, the following inequality holds: \
(2.84)

Let Pj be the centre of Bj, and denote by A2(a, D) the principal part of A{x,D): {A2{x,D)ipY=

i

a\,k{x)D,D^cp\x).

We denote by A2iPj, D) the partial differential operator with constant coefficients obtained from A2{x, D) by replacing a\iu{x) by their values at x=pj:

{A2{pj,D)
i

a\,^{pj)D,D^cp'{x).

i,/c = l

Applying Lemma 2.7 to their operator, we can take a positive constant C2{pj) depending on n, /JL, k, 6, 8 and MQ such that \
(2.85)

418

Appendix. Elliptic Partial Differential Operators on a Manifold

Since {{A2{x,D)-A,{pj,D))
i

{aUx)-aUpj))D,D,
i,k = l

by Lemma 2.11, we get a positive constant Q such that \A2{x, D)-A2(pj,

D))
(2.86)

Ci depends only on n, /JL, k, and 0. From (2.85) and (2.86) we obtain \
D)(pj\k+0-\-\(pj\k

+

CiC2(Pj)s^Mj,+e\
Since for a sufficiently small e C,C2{Pj)e'M^^e<2-\ we have 2+fc+0 = 2C2(/?;)(|A2(x, D)(pj\^^0-¥\(pj\j^).

(2.87)

In the same way as in the proof of Theorem 2.1, we see that there is a positive constant C^ such that

Hence |(p,-|2+fc+e^2C2(p,)(|A(x,D)(^,|fc+,4-|(p^4) + 2C2(;7,)C3|(p,|/c+i+.. Since [A(x, D ) , coj] is a first-order differential operator, there exists a positive constant C4 such that |[A(x, D ] , co^](p|fc+0 ^ C4Mk+0\(p\k+i+e' By the way, there is a positive constant C5 such that \(Pj\k ^ Csl^lfc

and

\(pj\k+i+e ^ d(p|fc+i+^.

Therefore J~^\(p\k+2+e ^2C2(Pj){\A(x,

D)
+ 2C2(/7,)C3C5kU+i^,.

(2.88)

By Lemma 2.8, we can find a positive constant C^ such that 2C2(/?,)(C4M;,+ , + C3C5)|9k+l + . + 2 C 2 ( / ? , ) C 5 | ^ k ^ 2 - V - ^ | ( p | , ^ 2 + e + Q k l 0 .

§3. Function Space of Sections of a Vector Bundle

419

Therefore by (2.86) 2-'j-'\
I

§3. Function Space of Sections of a Vector Bundle Lemma 3.1. Let U and V be domains in T" which are diffeomorphic to each other, and ^: U ^ V the diffeomorphism between them. Let K be an arbitrary compact set in U. Put K' = ^(K). Then for any integer I, there exists a positive constant C depending on /, 0 and K such that for fe C°^{V,C^) with supp f^K\

C-II/IINII/-CDIINCII/II?.

(3.1)

Proof For / ^ O , the lemma follows immediately from Proposition 1.2. In the case of / < 0, we set l = -m. Let Co be a constant with which (3.1) holds for / = m. Since supp/ci K\ f can be extended to a C°° function T" which is identically zero outside of V. Choose a C°° function x with support in V which is equal to 1 on K'. Then for any e C°'(T", C^), {L
SUp-T

\W\\m ^Ci

sup

—

cp lU^llm .

(3.2)

^ II^AIIm Here (A runs over C^iV, C'')\{0}. Let J{x) be the Jacobian of ^ and put 4f{x) = J{x){ilj'^){x). Then {f il/) = {f'^,ji). From Proposition 1.7, we obtain a positive constant C2 such that ||i^||^ = C2||«A'^lU- By (3.1) for l=m, we have \\iP'^\\m^jQ>MmSince for (/.e C^(y, C^)\{0}, i^e c?^(i/,c^)\{0},

^ ^ P i r 7 ¥ ~ = ^ ^ ^ o ) ^ ^ P 11/ rT^il ^

II^AlIm

^

^^0C2\\f'

II^A-^^llm

n-m>

^WQ)C2SUp—-T ^

\m\m

(3.3)

420

Appendix. Elliptic Partial Differential Operators on a Manifold

Hence by (3.2) and (3.3), we have the first inequality in (3.1): ||/||_^CIVQC2||/-^||_.

Using O"^ instead of O, we obtain also the second inequality in (3.1). I By use of Proposition 1.5, we can prove (3.1) for any real /. But we do not need this fact here, hence we omit the proof. By using Lemma 3.1, the Sobolev space of sections of a vector bundle can be defined. Let X be an n-dimensional compact manifold, and X = VJj=i Xj an open covering of X by a system of sufficiently small coordinate neighbourhoods. Set xy. p-^ Xj(p) = (X](p),..., Xj(p)) be C°° local coordinates on Xj. This maps Xj diffeomorphically onto the open unit ball of R". Since the interior of the unit ball is diffeomorphic to a domain in an n-dimensional torus T", we may consider Xj as the diffeomorphism of Xj onto a domain Vj in T". Let (B, X, t) be a complex vector bundle over X. The following discussion is also true for real vector bundles. Let /JL be the dimension of fibre of B. We may assume that 7r~^(Xj) = Xj xC^ on each coordinate neighbourhood Xj. Let i/^ be a section of B over Xj. Then ijj can be identified with a C^-valued function il^jiXj) defined on the domain Vj in T", and it is written as follows: iPjiXj) = (il^jiXj),...,

il^^iXj)),

Xj G Vj.

(3.4)

Suppose that i/^ is defined on X^, too. Then we have a C^-valued function M^k), with XfcG VJ,: Mxk) = (il^i(xk),..., For peXjnXk relation:

^«Xfc)),

Xfc G Vk.

(3.4)'

with Xj = Xj(p) and x^^Xkip),

we have the transition

il^Hxj(p))=lb^,,(p)rk(Xk(p)),

(3.5)

p

where b%p{p) is a C°° function of ;? G XJ n X^. We denote by C°°(X, B) the set of all C°° sections of B over X. Choose C°° functions Xj{x) on X such that Y.xMf^h

supvXj^Xj.

(3.6)

For any C°° section ijj of B, define the product Xj^ by XJ^{X) = XJ{X)IIJ{X). Then we can consider it as a C°°-valued function on Vj, since supp Xj^ ^ ^y Moreover its support is contained in a compact subset of Vj. Therefore for an integer /, we can define the norm ||A}
421

§3. Function Space of Sections of a Vector Bundle

means that Xj is identified with V, by Xj. Using this, we define the /-norm 11/ on X as follows:

'•=UxM\li=UxrxJ'x}{xj)\\l,.

(3.7)

This norm depends on the choice of {Xj]j. Instead of {XjYj take {(OjYj such that X (Oj{xf = 1,

SUpp 0)j C= Xj.

j

Then the norm X^. || coyiAlll/ is equivalent to (3.7). To show this fact, it suffices to see that there exists a positive constant C independent of iff such that, for any «Ae C°°(X, J5),

C^'lWxjcil^hi^l

||a;,(A||,,/^CZ ||;t..A|lM-

(3.8)

For any k, we get, by (3.6), \^Mj,i

(3.9)

= Z Xk(Ojil/ J,l

k

We denote (xl(^j)ixj \y)) = ai^kj)(y),y^ Vj- Using the representation (3.4) of i/f on Vj, we have \\XkO)jO)\\jj =

\\a(^kj)^j\\jj.

(3.10)

The support of xl^^j is contained in Xj n X^ so that we can apply Lemma 3.1 to ^ = Xj-Xk\ For zeVk we put a^kjy ^iz) = d(^kj)iz)il/j'^ = Xj{z) respectively. Then by Lemma 3.1 we can take a positive Q such that

Putting ij/jiz) = (ij/jiz),...,

iAf (z)), we have, by the transition formula (3.5),

^'(^)=I^;fcp(^r(^))^?(2:). p

For bfkp is a C°° function, there is a positive C2 such that ||a(fc,-)(A7llM=^ll^(/c/)^/c|lMSince a(k,)(z) = a(fc,) • ^(z) = (xio)j){Xk\z)),

(3-12)

we have

\\a(^kj)^k\\k,i = \\Xk(Ojil/\\k,i'

(3.13)

Appendix. Elliptic Partial Differential Operators on a Manifold

422

Since Xk^j is a C°° function, there is a positive C3 such that \\xlcoj
(3.14)

By (3.10)-(3.14), we get \\xla^jil^\y^C,C2C,\\xM,j. By (3.9) and the above inequality, \\cojilj\\j,i^C,C2C,l\\xicH\,,ik

Taking the summation over j , we obtain l\\cojilj\\jj^C,C2C,Jl\\xkil^U,i k

J

which shows the right-half of (3.8). By the way, the roles of {cOj} and {xk} are symrrietrical. Therefore the left-half of (3.8) also holds. Hereafter, choosing {Xj} once and for all, we fix the norm as in (3.7). Clearly ||

For any t>0 and any ifj e C°°(X, B), the following inequality holds

ll'/'li''^7rf'"'"'"'""ll'^li'

(2°)

Foranyil,eC°°iX,B),

Y-n/a-v^^o-''mi-n_

(3.16)

Proof. For any 7, by Lemma 1.1, we have

for any ^ > 0 and any il/e C°°(X, B). Summing these with respect to 7, we obtain (3.15). Taking the minimum with respect to r, we get (3.16). I

§3. Function Spdce of Sections of a Vector Bundle

423

A linear operator A: C°°(X, B)-» C°°(X, B) is said to be a multiplication operator, if it is represented in the form of (3.4) as {M)^{xj{p))=laUxjip))iljfixjip)\

peXj,

(3.17)

p

where a^p{Xj{p)) are C°° functions and for fixed/)eXj the (^t X/x)- matrix (aj^p(xy(p))^A=i is a linear transformatin of the fibre Tt~^{p) at /?. Therefore A is considered as a section of the vector bundle Hom c ( ^ ) = B®B'^ where B* is the dual vector bundle of B, We put M =Z I I j p,\

sup |D^^.aj^,(x,)|. m^lxjeVj

Lemma 3.2. Let I be an integer. Then for a multiplication operator A there exists a positive C such that for any (p e C°°(X, B), ||A(p||,^CM|„||(p||;.

(3.18)

C depends on n, /, /JL, but is independent ofcp and representations of A. Proof Take {xj} as in (3.6). By (3.17) we have Xj^ = ^Xp

P

Since a^p are C°° functions, we can apply to A the case m = 0 in Proposition 1.4 and we see that there is a positive C such that \\XjM\\j,i

=

^^\i\\\xj
Squaring both sides, and summing with respect to j , we get the desired inequality by taking the square roots of both sides. I Corollary. In the identity (3.17), if there is such a positive 8 that inf min |det(a|p(x,))| ^ 8, j

(3.19)

XjeVj

then, for each /, we can take a positive C depending on /, 8, /JL, n and M^^, such that \m,^C\\Ail,\\,

(3.20)

for any

424

Appendix. Elliptic Partial Differential Operators on a Manifold

A linear map A: C°°(X, B)-> C°°(X, B) is said to be a linear partial differential operator of order m, if it has the form (Ail^)^{xjip))=

I

laf,^ixjip))D^il;^{xjip))

|Qf|^m

(3.21)

p

in the local representation (3.4), where afp^ are C°° functions and for a multi-index a = (a^, a2,..., a^) Df means »;=Dri---«a=(^)"'---(^)-.

(3.22)

Note that the meaning of Dj above is somewhat different from the one given in §1. We put for any non-negative integer / M, = Z I I j a p,\

I

sup \Dfaf,^ixj)\.

(3.23)

iPl^lxjeVj

Lemma 3.3. Let A be a linear partial differential operator which operates on C°°(X, B). Then there exists a positive C such that for any (p e C°°(X, B) the following inequality holds.

||A(^||,^CM|,| 11(^11;^^,

(3.24)

where C is determined by /, m, n and /n and independent of representations of A. Proof Take Xj ^s in (3.6). Fix j , and let a)^, /c = 1 , . . . , / , be C°° function on X with supp cofcc: Xfc and Xk ^fc(^)^= 1 such that (Oj{x) = 1 identically on a neighbourhood of the support of Xj- Such {cok} do exist. Since coj is identically equal to 1 on the support of Xp Xj^^ — Xj^^j^- Hence \\XjA(p Wjj = WxjAcojcp Wjj ^ \\Axj(p y + WlXj, A]o)j(p\\jj. By Proposition 1.4 and Lemma 1.4, we have a positive Q such that \\^Xj(p\\j,i = CiMm ||^^-
(3.26)

and ||[^^-, A]a)j(p\\jj^CiM^ii \\a)j(p\\jj+rn-i'

(3.27)

By the choice of {(o^}, (3.8) holds. That is, there is a positive C2J such that Z \\(Oj,(p\\kj+rn-l=C2jY. \\Xk
k

(3.25)

§3. Function Space of Sections of a Vector Bundle

425

By (3.25), (3.26), and (3.27), IIA'J^^^IL/^

CiM\J\\Xj(p\\j,l+m-^C2j I \\Xk
Summing with respect to j and putting C3 = Xj Q j , we have WXjiphl+m-^CsY^ \\Xk(p\\k,l+m-lj

Z \\Xj^^\\jJ=CiMi^\T

^C,M^i^(l + C,)l\\xk
From this inequality, it follows

j

k

That is,

This proves (3.24).

I

For two C°° sections (p and J/^ of a vector bundle B, the inner product {(p, il/)i of order / is defined as (
(3.28)

j

Here the inner product of the right-hand side is understood to be the one of the Sobolev space, for, the supports of Xj

\\l = {
(3.29)

The completion of C°°(X, B) with respect to the norm || \\i is called the Sobolev space of order /, denoted by W\X,B). The inner product (3.28) extends canonically to the one on W\X, B), for which we use the same notation in (3.28). W\X, B) becomes a Hilbert space with respect to this inner product. Clearly this inner product depends on the choice of {Xj}But the norms determined by {Xj} are all equivalent. Therefore, as a vector space, W\X, B) is independent of the choice oi{Xj}, and so is its topological structure.

426

Appendix. Elliptic Partial Differential Operators on a Manifold

By Lemma 3.3, a linear partial differential operator A can be uniquely extended to the continuous map from W\X, B) to W^'^^iX, B). In particular, a multiplication operator is a continuous map from W\X, B) to W\X, B). For S 6 W\X, B), the support of S denoted by supp S is defined as the smallest compact set Ka X such that the product aS vanishes for any C°° function a(x) whose support is contained in X\K. If S is an arbitrary element of W\X, B) and xM is a C°° function, then supp X' S'^ supp X ^supp S.

(3.30)

Suppose that the support of x is contained in a coordinate neighbourhood Xi and take a family of C°° functions {OJJ} on X with supp o;^ c: Xj and X^. 6jfy(x)^= 1 such that coi(x) = 1 on supp x- Since Se W\X, B), we have a sequence {(pk}°k=^i in C°°(X, B) which converges to S in W^(X, B). Since the norms are equivalent if we take {(DJ) instead of {Xj}, there is a positive Ci such that for any /c, /c'

llAr
§3. Function Space of Sections of a Vector Bundle

427

putting /c->oo, we get

\\s,\\l,sclcl\\s\\^sclcl^\\Sj\\l,. j

Thus there exists a constant C such that, for any Se W\X, C-'l\\Sj\\li^\\S\\^^Cl\\S\\l j

B), (3.31)

j

Theorem 3.1 (Rellich's Theorem). Let V < I. Then W\X^ B) is contained in ^ ( x , B), ^^d the inclusion map W\X, B) -^ W^'(X, B) is a compact linear map. Proof. It suffices to show that the inclusion map is compact. Let {5^}^=i be a sequence in W\X, B) such that there is a positive constant R with

\\S%
L W^j \\j,l=J<

corresponding to j =

»

j

in other words, for each j , {5}}^=i is a bounded sequence in Wl)(Vj,C^). Thus since it is a bounded sequence in W^^(T", C^), we can choose a subsequence convergent in W^'(J'',C^) (Theorem 1.10). We may assume that these subsequences have the same set of indices for all / We write them as {Sf^}. We can choose a sufficiently large KQ such that for any K, K'> KQ, and for any 7,

\\Sf-Sn\j,v<J-'C-'e holds, where C is the same as in (3.31). Therefore, by (3.31), for K, K'> KQ, we have

||5^-S'^l|,<e. Thus {S^}K forms a Cauchy sequence in W^'(X, B), converges in W^'{X, B). Therefore the map W\X, B)-^ W\X, B) is compact. I Let t/^ be a section of B over X, and ilfj{Xj) the vector-valued function (3.4) of Xj on Vj, corresponding to the restriction of ifj to Xj. ij/ is called a C^^^ section of B if for each j , ^j{Xj) is a C^^^ function of Xje V), where 0 ^ (9 < 1. The set of all such ifj is denoted by C^^^X, B). We take {Xj] as in (3.6) and define the norm of C^"'^(X, B) by \^\u^e=l\xMj.k^e.

(3.32)

428

Appendix. Elliptic Partial Differential Operators on a Manifold

where | 1^;^+^ means the norm of Xj^ as an element of C^^^{Vj, C^). For a different choice of {;^^}, (3.32) defines an equivalent norm, because a similar estimate to (3.8) holds for | \ji+o. Theorem 3.2 (Sobolev's Imbedding Theorem). Let I and k be non-negative integers, and suppose l> n/2 + k.JTienforfe W\X, B), there exists a unique feC^{X,B) such that f(p)=f{p) on X except for a set of measure 0. Furthermore, there exists a positive constant C depending only on k and I such that |/|fc^C||/||,

(Sobolev's inequality)

(3.33)

holds. Proof Both W\X, B) and C^(X, B) are imbedded continuously into W^(X, 5 ) , and C°°(X, B) is dense in W\X, B). Therefore it suffices to show the equality (3.33) for fe C°°(X, B). In this case f{p) =f(p) for all peX. By (1.23) in the proof of Theorem 1.3, there is a positive constant Q such that \Xjf\j,,SCj\\x/h,,

(3.34)

Putting max C, = C, and summing (3.34) with respect to j , we have

I/U^CI;||A;/IL,^CV7||/||,, J

which proves (3.33). I Assume that X is oriented. By a volume element of X, we mean a C°° differential n-form v(dx) which is positive at every point of X. If a subset iV of X is of measure 0 with respect to one volume element, so is it with respect to another one, hence, the notion of the sets of measure 0 does not depend on the choice of a volume element. By a metric g on a vector bundle B, we mean a C°° section of the vector bundle B*®B* such that the following condition is satisfied: Let pe V), and let {p, ^), {p,C') be two points on the fibre 7r~^(p) with fibre coordinates {, = U],..., ^n, and ^; = Uj\ . . . , Cn respectively. Then the Hermitian form on Tr'^ip) defined by

gM.n=lgi.,Mp))iH^' A,P

is positive definite.

(3.35)

§3. Function Space of Sections of a Vector Bundle

429

Let (p,il/eC°^(X,B). In terms of fibre coordinates over Xj, we write ^iip) = {
=

^{p))v{dx)

I giKp{Xi{p))ip''i{x,{p))ri{Xi{p))v,{x,{p))

dx,

(3.36)

J X KP

This has an intrinsic meaning. With this inner product, C°°(X, B) becomes a pre-Hilbert space. The completion with respect to the norm

lkll = (^,
(3.37)

is a Hilbert space, which we denote by L^(X, B). This is the same as W^(X, B) as a topological vector space, but with a different inner product. Proposition 3.2. We fix a metric g and a volume element v(dx). The inner product ( , ) defined on C^{X, B) is uniquely extended to a continuous sesquilinear map of W~\X, B) x W\X, B) to C for any integer /, which we denote by the same ( , ). This is non-degenerate. Proof Take {Xj} as in (3.6). Then for (p,il/e C°°(X, B), we have i
(3.38)

j

Since for each j , (Xj^,Xj^)=

I gjxp(Xj(p))Xj(pf(^j(P))Xj^j(Xj(p))vj(xj(p)) J X KP

dXj(p)

(3.39) by Theorem 1.11 in §1, for any integer /, there is a positive constant C,j such that \{Xj, xM

g Qj\\xj
\\xM\j,i-

Summing with respect to j , and using Schwartz' inequality we have \{q>,il>)\^C,\\
(3.40)

where we put C; = max^ Cy. Thus ( , ) is continuous with respect to the relative topology of C^iX, B) x C°°{X, B) in W-\X, B) x W\X, B). Since

430

Appendix. Elliptic Partial Difierential Operators on a Manifold

C°°(X, B) X C°°(X, B) is dense in W-'(X, B) x W'(X, B),( , ) is extended uniquely to a sesquilinear map of W~'{X, B)xW'{X,B). Let 5 e W"'(X, B). Then xjS is identified with S, e WQ'(Vj, B). Define r^ = ( T ] , . . . , r j ' ) b y T"; = I Vjixj)gj,,{x)Sl

p = 1 , . . . , At.

(3.41)

A

Then by (3.38) and (3.39), we have (S.
(3.42)

j

where ( , ) is the natural bilinear map of Wo\Vj, C'') x WliVj^C"} to C on Vj. The left-hand side is iiidependent of the choice of{xj}. Suppose that Se W-\X, B) satisfies (5, (^) = 0 for any cp e C°°(X, B). For an arbitrary a e C°°(X) with supp a c: X^, there is a set {Xj}j=i satisfying (3.6) such that Xk(x)=l on supp a. Since (3.42) holds for such {xj}, we have for any
j

Thus 5 = 0, which implies the non-degeneracy of ( , ). I

§4. Elliptic Linear Partial Differential Operators (A) Estimates with Respect to the Sobolev Norm (a) A priori Estimate Let X be a compact manifold of dimension n, B a complex vector bundle over X with /x-dimensional fibres, and TT: B^X its projection. We use the same notation as in §3. Let A(/7, D): C°°(X, B)->C°°(X, B) be a linear partial difierential operator of order /. In terms of the local coordinates xy. Xj-^ VJ on the open

§4. Elliptic Linear Partial Differential Operators

431

set Xj, a section cp e C°°(X, B) of B is written as
cl>^ixj{p))),

We write the operation of A{p, D) as I Y^aU{xj{p))DJ
{A{p,D)cp)^{xj{p))=

where the a^pai^jip)) are C°° functions of Xj{p). The operator consisting of the terms with |a| = /, given by {A,{p.D)^)^{xj{p))=

I la%^{xj{p))DJ
(4.1) Ai{p,D)

(4.2)

\a\ = l p

is called the principal part of A{p, D). Let f{p) be a real-valued C°° function on X, and suppose that ^p = df(p)7^0 at /?. Then r ^ e'^^^p^A{p, P)e"'^^^^V(p)) is a polynomial of degree / in t, whose coefficient of the terni of degree / is given by iA(p, iQ
'
(4.3)

\a\ = l p

where we put ^p = df{p) = ^j^ dx] + • • • + ^^„ dxj. For any non-zero element ^p in the cotangent space r*X, there is an fip) with fp = df{p). For any element w in the fibre Bp at /? G X, there is a section (p such that (^(p) = w. Let ^pGT*X be given by ^pj = ^ji
I Iaj;,«(x,(;7))^;w;,

A = 1,...,M

(4.4)

\cx\ = l p

for 0 # ^p G T'pX, which we call r/ie principal symbol of A(/7, D). Definition 4.1. A(/7, D) is said to be elliptic if for any/? e X, and any fp e T^X with ^p 7^ 0, the principal symbol Ai{p, ^p) is an isomorphism of Bp = 7r~^(/?). In this case, there exists a positive constant 6 such that for any ^p e r * X with fp 7^ 0, and w e Bp, the inequaHty l[iAm{p,$,)w)^f^8'l\wf\\ A

(4.5)

A

The supremum of such 8 is called the constant of ellipticity ofA{p,

D).

432

Appendix. Elliptic Partial Differential Operators on a Manifold

We introduce the following constants as in the preceding section. For /c = 0 , l , . . . , put Mfc = I I Z j

I

a p,\

sup \Dfa^,^{xj)\.

(4.6)

|j8|gfc XjVj

Theorem 4.1 (L^ a priori Estimate). Let A(p, D) be an elliptic linear partial differential operator of order I For any integer k, there is a positive constant C depending only on w, /, k, /x, the constant d of ellipticity of A{p, D) and M|fc|, such that for any u e W^^^{X^ B), the inequality ||M|U^,^C(||A(/?,D)M|U+||I/|U)

(4.7)

holds. Proof Since both sides of (4.7) are continuous with respect to the topology of W^-^'iX, B), and C°'(X, B) is dense in W^-^^X, B), it suffices to prove (4.7) for any u e C°°(X, B). Take functions {Xj}j=i as in (3.6) in the preceding section. For any u e C°°(X, B), there is a j such that r^^u\\,^,^\\Xju\y^i.

(4.8)

Put Uj = XjU. Take another family of functions {wi}f=i on X such that supp a)j<^ Xj and that X, ^iip)^^ 1 ^^ ^' Moreover we assume that Wj is identically equal to 1 in some neighbourhood of supp Xj- Then we have \\A(p, D)uU^\\XjA{p,

D)u\y

= \\XjAip,D)coju\y = | | A ( A D)uj\y-\\[A(p,

(4.9) D), iOj]coju\y.

Since [A{p, D),Xj] is a partial differential operator of order ( / - I ) , by Theorem 1.4, there is a positive constant Q such that \\[A(p, D), Xj](oM\j,k^ C^M\k\\\(Oju\\j^k+i-i,

(4.10)

while there is a positive constant C2 by (3.8) such that ||^yw|b,k+/-i^ZlkiW||,;fc+/_i^C2||M|U+/-i.

(4.11)

i

Since the support of Uj = XjU is contained in the coordinate neighbourhood Xj, Uj can be considered via the local coordinates Xj as an element of C°°(T", C^) with support in Vj. Thus we can apply Theorem 2.1 to find a positive constant C3 depending on k, /, n, /JL, d and M|fc| such that ||A(/7,D)M,-||,.,;,^C3||M,-||,-,;,+ /-||M,-Lfc.

(4.12)

§4. Elliptic Linear Partial Differential Operators

433

From the inequality || w, 11^,^ ^ || w |U, and (4.8), (4.9), (4.10), (4.11) and (4.12), we obtain ||A(AI^)w|U^C3J^/'i|t^lU-^/-||w|U-C2QM|„||u|U^;_,.

(4.13)

By Proposition 3.1, there exists a positive constant C4 such that

hence from this and (4.13), we obtain ||A(p,D)i/|U + (C4+l)||M|U^2-^C3/^/^i|w|U,z.

I

(b) Garding's Inequality As stated at the end of §3, we consider the Hilbert space L^(X, B) defined by the volume element v(dx) of X and the metric g on the vector bundle B. We denote its inner product by ( , ). Lemma 4.1. Let A{p, D) he a linear partial differential operator of order (m-\-l). Then there exists a positive constant C determined by m, n and JJL such that for any cp, il/ ^ C°°(X, B), \iAip, D)ip, 4f)\^CM,MJ^\\,

(4.14)

holds. Proof. Take {xj} as in (3.6). Then we have (Aip, D)
(4.15)

J

Fix7, and choose a family of C°° function {ifj^Yk^i with supp co^^ Xp, and X o)\{x) = 1 so that (Oj is identically equal to 1 in some neighbourhood of supp Xj' Then we have (A(/7, D)(p, x]^) = (A(p, D)a>j
I

Z

gjK-MiPm.M(p))

+ / p,A,o- J V

xDf{cojl^])(.Xjip))vjixj{p))dXjip).

434

Appendix. Elliptic Partial Differential Operators on a Manifold

Integrating the terms with |a| ^ m + 1 by parts {\a\-m) (A(p,D)coj
I

Z

[

times, we have

bj,,^,(xjip))DJ(coj
\cx\^m p,\,cr J Vj

xDf{xjiPf)ixj{p))vj(xj(p))

dxjip),^

where the bj^pa/iiXjip)) are Hnear combinations of partial derivatives of Xjgj\p(Xj(p))aj^^{Xj(p))vjixj{p)) of order at most /. Therefore there exists a positive constant Q such that

By (3.8), there is a positive constant C2J depending on 7 such that \(A(p,D)
l\{A{p,D)cp,xjH^C,M,\\
(4.14) follows from this and (4.15).

I

Let A(p, D) be a linear partial differential operator of order 2m. For peX, and ^p e TpX with ^p 9^ 0, the principal symbol A2mip, ^p) of A{p, D) is a linear map Bp -^ Bp. Therefore, by virtue of the metric gp on Bp, we can define a Hermitian form on Bp x Bp associated with this linear map by putting gpiA2m{p,Qw,w')

(4.16)

for we Bp and w'e Bp. For p e Xj, if we write A{p, D) in the form of (4.1) with / = 2m, then putting ^p = ^ji dx] + ••• + !;„ dx], iv = ( w j , . . . , w"), w' = (wj\ . . . , wj"), we have

cr,\,p |a| = 2m

Definition 4.2. A linear partial differential operator A(p, D) of order 2m is said to be strongly elliptic if there exists a positive constant 8 such that for any peX, any ^p e T*X with ^p 9^ 0, and any weBp with w 7^ 0, ( - 1 ) - Re gp{A2mip. ^p)w, w) ^ 8'gp(w, w)|^,|^-

(4.18)

holds. The supremum of such 5 is called the constant of strong ellipticity of Aip, D).

§4. Elliptic Linear Partial Differential Operators

435

Theorem 4.2 (Garding's Inequality). Let A(p, D) he a strongly elliptic partial differential operator of order 2m. Then there are positive constants 8i, 82 depending on m, n, ^c, the constant 8 of strong ellipticity ofA(p, D) and M^ such that for any cp e C°°(X, B), Re(A(p,D)
(4.19)

holds. Proof Take {Xj}j=i as in (3.6). For simplicity we write A for A(p, D). Since lLjXj{sY=^. we have {Acp, (p) = Y. (A(p, Xj
j

= E (^Xj(p. Xj
j

Since A}[A}, ^ ] is a linear partial differential operator of order ( 2 m - 1 ) whose coefficients are linear combinations of those of A{p, D), by Lemma 4.1, there exists a positive constant Cj depending on n, m, /x and M^ such that \{Xj[Xj,A]
From the above inequalities we obtain

R^{A
j

= C3||
(4.20)

By Proposition 1.1, there is a positive number C4 such that

Consequently by (4.20) above, we obtain Re{Acp,
436

Appendix. Elliptic Partial Differential Operators on a Manifold

we obtain Rc(Acp,cp)^'2Cs\\cp\\l-Cs(C2+Cd{
I

(B) A priori Estimate with Respect to the Holder Norm Lemma 4.2. Let 0
Then there exist positive constants Cj, C2 such that \
and that for any t>0 \
I

Definition 4.3. Let A{p, D) be a second-order linear partial differential operator acting on sections of a vector bundle B over X. The principal part of A{p^ D) is said to he of diagonal type if we can choose a system of local coordinates {{Xj, X/)}/=i such that when we write A(/?, D) in terms of these coordinates as {A{p,D)cj>)^{xj{p))=

E Za;,Jx,(p))Z>;<(x,-(i7)),

(4.21)

then the following conditions is satisfied: «ia(^j(;?)) = 0

for |a| = 2

and

A 7^ p,

P^Xj.

(4.22)

Remark. If in some choice of local coordinates, the principal part of A(/7, D) is of diagonal type, and it is written as {A2{p,D)ip)^{xj{p))=

I

ajAxj{p))D^cpf(xj(p)),

(4.23)

l«H2

then the principal part is of diagonal type in another choice of coordinates. Theorem 4.3. Let A(p, D) he a strongly elliptic partial differential operator of order 2 with C°° coefficients acting on sections of a vector bundle B over X whose principal part is of diagonal type. Let 8 be its constant of strong ellipticity, and put 0< 6 <1. Then for any integer k^O, there exists a positive constant C depending only on 0, k, n, /x, 8 andMj^-^e such that for any u e C^^^'^^(X, J5), lwL^-2+a ^ C{\A{p, D)u\k^e + Nlo)

§4. Elliptic Linear Partial Differential Operators

437

holds, where we put Mfc+^ = Mfc + X S

I

sup

^^—

j \a\^2 \p\^k \x-x'\^l

-e

.

\X — X\

Proof. We may assume that the system of local coordinates {^}/==i is so chosen that the principal part is already diagonal. Take {Xj}j=i as in (3.6). For u e C°°(X, B), by the definition of the norm (3.32), there is 3.j such that J~^\u\k+2+e ^ \xMj,k+2+e'

(4.24)

Put XjU = Uj- Choose a family of C°^ function {<w,}f=i on X with supp (Ot c: X, and Xi (^iip)^= 1 such that (Oj(p)= 1 on some neighbourhood of supp;^/. Then we have \A(p, D)u\k^e ^ \XjMp, D)u\j^j,+e = \XjMp. D)(OjUy+e ^ \A{p, D)uj\j,,,^e - |[A( A D), xMAu^^e^

(4-25)

Since [A(/7, D), ;^^] is a first-order partial differential operator, by Lemma 2.8 there is a positive constant Cj depending only on fc, ^, n, /x such that |[A(/7, D), A}]w,wb,/c+0 ^ CiMfc_,^|w^M|,;fc+n.^,

(4.26)

while there is a positive constant C2 such that |w^M|^;fc+l + 0 ^ C2|M|fc+i + 0.

(4.27)

Since the support of w^ is contained in X^, M, can be considered, via x^, an element of c^''^''^(T^ C") with support in V;. Hence by Theorem 2.3, there is a positive constant C3 depending only on /c, ^, w, /x, 5 and Mk+g such that C3|w,b;fc+2+a - I W J U O ^ | A ( A D)M,^,fc+,.

From thQ trivial inequality (4.28), we obtain

|M,|,;O^|W|O

(4.28)

and (4.24), (4.25), (4.26), (4.27),

|A(AD)wU^,^C3J~'|M|k+2+.-|w|o-CiC2Mfc^a|wk+i+a. By Lemma 4.2, there is a positive constant C4 such that CiC2Mfc+^|M|fc+l + ^ ^ 2 ~ ' C 3 / ~ ' | M | f c + 2 + ^ + C 4 | M | o .

(4.29)

438

Appendix. Elliptic Partial Differential Operators on a Manifold

Consequently from this and (4.29), we obtain

which proves the theorem.

|

Remark. Let A(x, D) be an elliptic operator of order / with C ^ coefficients, and 8 its constant of ellipticity. Then if 0 < ^ < 1 , for any integer fc^O, there is a positive constant C depending on n, /, /JL, k, 6, 5, Mk+e such that \u\j,^j^e ^ C(|A(x, D)M|fc+, + |M|O). We omit the proof.

§5. The Existence of Weak Solutions of a Strongly Elliptic Partial Differential Equation As was stated at the end of §3, we fix once for all a volume element v(dx) of X and a metric g on the vector bundle B, Then we can define L^{X, B) with respect to ( , ). Proposition 5.1. For a linear partial differential operator A{p, D) of order /, there exists a unique linear partial differential operator A{p, D)"^ such that for any (p^ijje C°^(X, B), the following equality holds: (A(A D)cp, (A) = (
(5.1)

Moreover A{p, D)"^ is also of order I. Proof Let {^j}/=i be a C°° partition of unity subordinate to the open covering \Jj=.^Xj of X Then we have {A{p, D)
If A(p, D) is written in the form (4.1) on Xj, we have

( A ( A D)COJ
p,\,(T

I

[

gj.,{xj{p))a)^^(xj{p))

J Yj

xDf{coj
4xj{p).

(5.2)

439

§5. Weak Solutions of a St;rongly Elliptic Partial Differential Equation

Since cojipj vanishes near the boundary of V), by partial integration, we obtain

v,k J V.

x4>]-{x^{p))vj{xj{p)) dxj{p),

(5.3)

where we put A,cr,p|a|S/

^MAP))

xD;(g,.,,(x,-(/7))<«(x,-(/7))i;,. (x,-(p))iA;(x,(/?))), (5.4)

and {gj'^ixjip))) is the inverse of the matrix {gjxp{Xj{p))). Therefore if we put tA' = Zj ^j^'j-> then A(/7, D)"^: ifj-^ifj' is the required differential operator. The uniqueness if obvious. I Definition 5.1. A(p, D)"^ is called the formal adjoint ofA{p, D). If A{p, D) is of order /, so is A(/?, D)"^. Proposition 5.2. Let peX and ^^ ^ T"^^ with ^^ T^ 0. If we write the principal symbol of the differential operator A{p, DY CLS Ai{p, ^x)"^, ^^^^ It is the adjoint of the linear map ofBp determined by the principal symbol Ai{ p^ ^^) ofA(p, D) with respect to the metric gp. Proof Let f{p) be a real-valued C°° function on X with df{p) = ^p. Then for any
D)e''^(p, lA), (5.5)

I

We want to solve the partial differential equation A{p,D)u{p)=f(p).

(5.6)

Definition 5.2. L e t / e W'^iX, B). u e W'^iX, B) is said to be a weak solution of the equation (5.6) if for any (p e C^iX, B), iu,Aip,Dr
(5.7)

holds, where A(p, D)"^ is the formal adjoint of A{p, D), and ( , ) denotes the extended one in Proposition 3.2. Proposition 5,3.Let A(p, D) be a partial differential operator of order I. Then

440

Appendix. Elliptic Partial Differential Operators on a Manifold

by Lemma 3.3, it is extended to a continuous map ofW^{X, B) to W^~^{X, B). If u is a weak solution of (5.6), in this sense, we have Aip,D)u=f

(5.8)

in W^-'(X, B). Proof Let {(p„}^=i be a sequence in C°°(X, B) converging to u in W^(X, B). Then {A(/?, D)
Hence A(/7, D ) M = / in \y'^-'(X, B),

I

In particular, i f / e C^(X, B), M is a weak solution of (5.6), and ue C\X, B), then by Proposition 5.1, for any cp e C°°(X, J3), we have {Aip,D)u,
(fcp\

hence A(p, D)u = / in the sense of L^(X, B). But since both sides of this equality are continuous, we obtain A{p,D)u(p)=f{p) for all p&X, namely, we obtain a solution of (5.6) in the usual sense. In order to show the existence of a weak solution, the following LaxMilgram theorem is useful. Theorem 5.1. Let ^ be a complex Hilbert space, ( , ) its inner product, and I I its norm. Suppose that B{x,y) is a Hermitian form on ^ satisfying the following condition: There exist positive constants Ci ^ C2 such that for any x,yeX, \B(x,y)\^C2\x\\y\, ReB(x,x)^Ci|x|^

(5.9) (5.10)

Then for any continuous conjugate linef{x) on S€, there exists a unique element Fsof^with B{Fs,z)=f(z),

(5.11)

Proof As will be shown later, there exists a continuous linear isomorphism

§5. Weak Solutions of a Strongly Elliptic Partial Differential Equation

B of ^ onto ^ such that for any

441

x,ye^

B{x,y) = (Bx,y)

(5.12)

holds. Assuming this for a moment, we see that by the Riesz representation theorem, there exists an element ye^ such that for any ze^ {y,z)=f{z). Hence it suffices to put FQ = B~^y. We shall prove the existence of B satisfying the above condition. Fix an arbitrary xe^. The associated linear map ^sy -^ B{x,y)eCis continuous by (5.9), hence, by the Riesz representation theorem, there exists a z{x) e ^ such that B(x, y) = (z(x), y) for any y. The map x -> z(x) is linear. Therefore there is a linear map B with z(x) = Bx. By (5.9) and (5.10), we have CAx\^

— 1^1

^\Bx\ = sup—:——^ y^o \y\

C2|x|.

(5.13)

Hence Bx is continuous. If Bx = 0, by the left inequality of (5.13) we have X = 0, hence B is injective. Furthermore the range of B is closed. In fact, if {BXn}n converges to y, then by the left inequality of (5.13), {x„}„ is also a Cauchy sequence, hence, converges to some z. Since B is continuous, Bz = y. Moreover the range of B is dense. For, if z is orthogonal to the range of B, we have 0 = \{Bz, z)\ = |B(z, z)| ^ Re B(z, z) ^ C,\z\^, which implies z = 0. Thus the range of B is dense. Consequently the range of B coincides with the whole ^ , and B is bijective. The continuity of B~^ is also clear. I Let A(/7, D) be a strongly elliptic linear partial differential operator of order 2m. For a sufficiently large A > 0, we shall prove the existence of a weak solution of (A{p,D) + XI)u = w.

(5.14)

For this, take {Xj}j=i ^^ i^ (3.6). Then with the inner product (3.28), W"(X, B) becomes a Hilbert space. Put (
= B{cp, (A)

(5.15)

for (p,il/e C°°(X, B), where the inner product in the left-hand side denotes that in L^(X, B).

442

Appendix. Elliptic Partial Differential Operators on a Manifold

Proposition 5.4. Let 8i, 82 be the positive constants given in Theorem 4.2. By (5.15), B((p, if/) defined on C°°(X, B) extends uniquely by continuity to a continuous Hermitian form on W"(X, B), which we denote by the same notation B((p, ijj). Then there exists a positive constant Cj determined by n, m, jx, and M^ such that 1/ Re A > 81, for any
(5.16) (5.17)

since B(^, (/f) = ((A(/?, D) + A)(p, f^), we have Proof If (p.il/eC^iX^B), (5.16) by Lemma 4.1. In other words, B((p, if/) is continuous with respect to the relative topology of C°°(X, B) in W^iX, B). Since C°°(A:, B) is dense in W"(X, B), B{(p, ifj) extends uniquely to a continuous Hermitian form on W""(X,B). (5.16) still holds for the extended B{(p,ip). By Garding's inequality, (5.17) holds for cp e C°°(X, B). Hence, by continuity, (5.17) holds for all cpeW'^iX^B). I Theorem 5.2. Let A(p, D) be a strongly elliptic linear partial differential operator of order 2m, and Sj, ^2 the positive constants given in Theorem 4.2. /f Re A > 81, for any w e L^(X, B), there exists a weak solution of the equation Aip,D)u-^Xu

=w

(5.18)

contained in W"^(X, B). Moreover the weak solution of this equation contained in ^"{X, B) is unique. Proof Let A{p, D)"^ be the formal adjoint of A{p, D), and B{(p, \fj) the Hermitian form on ^ ' " ( X , B) in Proposition 5.4. Since W^^X, B) is imbedded in L^(X, B), putting f{(p) = {w,(p) for any

(5.19)

where the left-hand side means B in the extended sense. We may assume that there are ifjk e C°°(X, B), /c = 1, 2 , . . . , such that {il/k}k converges to u in W'^iX, B). Then {(Afc}fc converges to u also in L^{X, B). Therefore, for

§6. Regularity of Weak Solutions of Elliptic Linear Partial Differential Equations

443

(p G C°°(X, B), we can write B{u, cp) = lim B((Afc, oo

= ( M , ( A ( p , D r + A)9).

(5.20)

Hence, w is a weak solution of (5.18). The uniqueness follows from the following inequality by putting w = 0. 0 = |(w, M)| ^ Re B{u, u) ^ 62IIM

12

§6. Regularity of Weak Solution of Elliptic Linear Partial Differential Equations Theorem 6.1. Let A(p, D) be an elliptic linear partial differential operator of order I with C°° coefficients. Suppose that for some integers s and k, fe W'~^~^{X, B). Then for a weak solution u e W'{X, B) of the equation (6.1)

A(p,D)u=f there exists a positive constant C such that \\ul)

^C{\\fU,.,+

(6.2)

holds, where C depends on /, n, k, /x, the constant of ellipticity 8 ofA(p, D) and M|5|, hut is independent of f and u. Proof First we prove the case k=l. Take C°° functions cOj, j = 1,..., X with cOj(p)^0 and supp coj c: Xj so that peX.

lcoj(p)^l,

J on

(6.3)

j

Put Uj = COjU,

fj = (Ojf

j=l,...,/.

Then in W'-\X, B), we have A(p, D)uj =fjHA{p,

D), coj]u,

7= 1,...,/,

while fj = (Ojfe W'~^^\X, B). Since [A(p, D), o)j] is a differential operator of order ( / - I ) , we have {A{p, D), o)j\ue W'~^'^\X, B). Furthermore, putting gj =fj-h[Aip, D), (Oj]u, we have gje W'-'^\X, B), and A{p,D)uj = gj.

(6.4)

444

Appendix. Elliptic Partial Differential Operators on a Manifold

Since the supports of Uj and gj are contained in Xj, Uj and gj can be identified with C^-valued distributions Uj e W'o( Vj, C^) and g, e W'o~'''\ V,, C^) on the open set Vj in T" respectively. The supports of these Uj and gj are contained in a compact set K. Writing w, = (M], . . . , Uj) and gj = {gj,..., g") in terms of their components, we have gj=

I

i:a^,Axj)Dy^

= {Aj{p,D)uj)\

xj = xj{p).

(6.5)

Since X is a compact set in V), there is a positive number /IQ such that if 0<\h\
/c - 1 , . . . , n,

(6.6)

where we put Aj;.g,= I

i(Aj;,aj^,j(x)D;TJ;,u;.

(6.7)

By Theorem 1.3, there exists a positive constant Cj independent of h such that ||i;^;fclb,5-z^ Qlkj;fcM;-||^> = CI||M,-||^;,.

(6.8)

For 0 < |/i| < /zo, we have ^luUj e Woi V;, C^). Since AJJ^w, satisfies (6.6), by Theorem 2.1, there exists a positive constant C2 independent of h such that ||Aj;,ii,-||,;,^C2(||5j;,g||,-,_,+||i;,;,||,;,_,+||Aj;fc^

(6.9)

On the other hand by Theorem 1.13, there is a positive constant C3 independent of h such that for 0<\h\< ho, the following inequalities hold: \\^j,kgj\\j,s-i^

C4gj\\j^,_i+i,

k=l,...,n,

\\^J,kUj\\j,s-l^Cs\\Uj\\j,s-M.

Using these inequalities and (6.8), we have the following estimate: For 0<|/2|< ho, and for any k = l,... ,n, ||AJ;fclI,.||,;,^C2C3(||g,-||,;._/+l+||lJ,-||,;._/+l)+QC2||^ Since the right-hand side does not depend on h, by Theorem 1.13(3°), we see that UJE W'O''\VJ,C^). Therefore we obtain UJE W'^\X,B). By the construction of Uj and (6.3), we have w =X Wy G W^^iX, B). Therefore by (6.1), using Theorem 4.1, we see that there exists a positive constant C4

§7. Elliptic Operators in the Hilbert Space L^(X, B)

445

such that ||w||,+l^C4(||g||._z+i+||ML-/).

Thus the theorem is proved for k=l. We proceed by induction on k>l. Suppose that the assertion is true for k^j. We want to show the case A:=j + 1. In this case we have fe W'-^-^'^'iX, B), and ueW'(X, B). By induction hypothesis, we have ue W^^^{X, B). Taking s' = s+j instead of 5, we see that (6.1) holds f o r / e W''-^'^\X, B), and ue W''{X, B). Since we have proved the theorem for /c = l, we have ueW^'^^{X,B) = W^^^^^{X, B). By Theorem 4.1, there is a positive constant Cs such that the following inequality holds:

Since this implies (6.2), we showed the case k=j + \, which completes the induction. I Corollary. Let A{p, D) be an elliptic linear partial differential operator of order /, and suppose that for fe C°°(X, B), u e W'(X, B) is a weak solution of A{p,D)u=f Then there exists v e C°°(X, B) such that in L^{X, B) u-v = 0 holds. Proof S i n c e / e C°°(X, B), for any integer A:> 0, we h a v e / e \V'"''^(X, B), hence, by the above theorem, we have ue W'^^'iX^B). If A:>[n/2]+l, by Sobolev's imbedding theorem (Theorem 3.2), there is a i;e ^,^-fc_c„/2]-i^^^ B) such that in W^(X, B) u = V

holds. Thus, u = v in L^(X, B). Since k is arbitrary, we have ve

C^(X,B).

I

§7. Elliptic Operators in the Hilbert Space L^(X, B) Let A{p, D) be an elliptic linear partial differential operator of order / with C°° coefficients. A{p, D) extends uniquely to a continuous linear map of W\X, B) to W'~\X, B). By restricting the domain of A(;7, D) appropriately, we treat A{p, D) as a closed operator of the Hilbert space L^(X, B).

446

Appendix. Elliptic Partial Differential Operators on a Manifold

(a) Closed Extension of Elliptic Partial Differential Operators Definition 7.1. We define a linear operator A in the Hilbert space L^{X, B) as follows. The doniain D(A) of A is given by D(A) = {ue L\X,

B) \ A{p, P)ue L\X,

B)},

(7.1)

and for u e D(A), we put Au = A{p, D)u. Theorem 7.1. (r)

D{A) = W\X, B), and the topology of D(A) defined by the graph norm coincides with that of W\X, B). (2°) A is a closed operator. (3°) C°°(X, B) is dense in D{A) with respect to the graph norm, that is, for u e D{A), there is a sequence {
(7.2)

while the middle part is just the graph norm of u. Thus (1°) is proved. (2°) Since D(A) endowed with the topology defined by the graph norm coincides with W^{X, B), this space is complete. Therefore the graph of A is a complete subspace of the direct product L^(X, B) xL^(X, B), hence closed. (3°) is clear from the density of C°°(X, B) in W\X, B). I Since the formal adjoint A{p, D)"^ of A(/7, D) is also a partial differential operator of order /, A{p, D)"^ has the unique extension to a continuous map of W\X,B) to W'-\X,B). By Proposition 5.2, Aip^D)"" is also elliptic. Theorem 7.2. Let A* be the adjoint of A in the sense of the operator on the Hilbert space L^{X,B). Then the domain DiA"^) is given by D{A*) = W'{X, B), and for v e D(A*), we have A*i; = A(;?,D)^i;.

(7.3)

In particular, if A(p, D) is a formally self-adjoint elliptic partial differential operator, A is self-adjoint.

§7. Elliptic Operators in the Hilbert Space L^(X, B)

447

Proof. V e D(A*) if and only if there exists an fe L^iX, B) such that for any u e D(A), {v,Au) = {f,u).

(7.4)

Since C°°(X, B) c D{A), for any

(7.5)

This implies that u is a weak solution of the equation

Aip,Drv=f. Since feL^(X,B), by Theorem 6.1, we have ve W^{X,B),. Conversely, suppose that v e W\X, B) and / = A(p, D)^v. Then fe L\X, B), and for any (p e C°°(X, B), (7.5) holds. Since both sides of (7.5) are continuous in (p with respect to the graph norm, and C°^{X, B) is dense in D{A) with respect to the graph norm, (7.4) holds for every u e D(A). Thus the theorem is proved. I Lemma 7.1. Let A(p, D) be an elliptic linear partial differential operator of order I. We define the operator A as in Definition 7.1. Then (1°) (2°)

ker A is a closed subspace of L?{X, B). For any integer /c ^ 0, there exists a positive constant Q such that for any u e W^'^\X, B) n (ker A)^, the following estimate holds: ||M||,^fc^Cfc||Aw|U.

(7.6)

Similarly if 0 < ^ < 1, for any integer k^O, there exists a positive constant Cj, such that for u e C^^^^\X, B) n (ker A)"", \u\i+k+e = Ck\Au\j,+e

(7.7)

holds. Proof ( r ) If {M„}^=I C: ker A, and if {M„} converges to u in L^(X, B), then AM„ = 0 converges to 0, and, since A is a closed operator, u G D{A) and Au = 0. This proves (1°). (2°) Suppose that for some /c, (7.6) does not hold for any Q . Then for any positive integer 7, there exists a WyG W^^^{X., B ) n ( k e r A)"^ such that ||M,-||/+fc = l

and \\AUj\\k^r\

(7.8)

Since the sequence {Uj}j is bounded in W^^^{X, B), by Theorem 3.1, {Uj}j is relatively compact in W^(X, B). Taking a subsequence, if necessary, we

448

Appendix. Elliptic Partial Differential Operators on a Manifold

may assume that {Uj}j converges to u in W^(X, B), while {AM,} converges to 0 in W^{X, B). In L^(X, B) also, {Uj}j and {Au^}j converges to u and 0 respectively. Since A is a closed operator on L^(X, B), we have u e D(A) n ker A On the other hand, since {M;}CI (ker A)-^, and (ker A)"^ is a closed subspace of L^(X, B), we have u e (ker A)"^. Hence M = 0 in L^(X, B). Since we have already known that u e W'^iX, B), we see that M = 0 in W^{X, B). Since {uj}j converges to u in W^{X, B), lim||M,-|U = 0.

(7.9)

Using the a priori estimate in Theorem 4.1 for {«/}<= W^'^'^{X, B), we see that there is a positive constant C such that ||M,.|U^,^C(||AM,-|U + ||M,.|U),

7 = 1,2,....

Taking the limits of both sides for j ^ o o , we have 1 ^ 0 by (7.8) and (7.9), which is a contradiction. Hence (7.6) is true. (7.7) is proved similarly. I Theorem 7.3. Let A(p, D) he an elliptic partial differential operator of order I. We define the operator A and its adjoint A* as in Definition 7.1. Then we have the following, (1°) (2°) {T)

Both ker A and ker A* are finite-dimensional subspaces of C°°(X,B). The range R{A) of A and the range R{A^) o/A* are closed subspaces ofL\X,B). R{A) = (ker A*)-', JR(A*) = (ker A)^.

Proof ( r ) By the Corollary to Theorem 6.1, we have ker A c C°°(X, B). Let Q = ker A is a Banach space as a subspace of L^{X,B). {wGker A | | | w | | ^ l } be its unit ball. Since Qc: C°°(X, B), by Theorem 4.1, we see that if M G Q , | | W | | / ^ C||M|| ^ C. Thus Q is bounded in W\X,B). Therefore by Theorem 3.1, it is compact in L^{X, B), hence in ker A. Consequently ker A is a locally compact space, hence finite dimensional. (2°) Suppose that a sequence {fj}t=i in R{A) converges t o / i n L^(X, B). By Lemma 7.1, for any j a n d / , we have the following inequality:

\\uj-urh^C,\\fj-ffl Since {fjj is a Cauchy sequence, {Uj}j becomes a Cauchy sequence in W\X,B), hence converges to some ue W^(X,B) in W^(X,B). On the other hand, AM„ =/„ converges to / in L^(X, B), hence we have Au =f Thus R{A) is a closed set. The closedness of RiA"^) can be proved similarly. (3°) Let ueR(A)-^. Since for any D G D ( A ) , (AU, M) = 0, we have ue D{A*), and A * M = 0 . Conversely if A*M = 0, clearly M 1 J R ( A ) , hence we

§7. Elliptic Operators in the Hilbert Space L^(X, B)

449

have ker A* = R{A)^. Since R(A) is closed by (2°), we obtain (ker A*)"- = R(A). (ker A)^ = RiA"^) is proved similarly. I Lemma 7.2. Let P and Q be the orthogonal projections to ker A and ker A* m L^(X, B) respectively. Then for any integer /c^O, there exists a positive constant Cj^ such that for any u e L^(X, B), the following estimates hold. \\PuU^a\\ul \Pu\j,^e^a\\ul

||QM|U^C,||I/||. \Qu\j,^e^a\\u\\.

(7.10) (7.11)

Proof. Let (fu - - -, (PL be an orthonormal basis of ker A. Then Pu = Hj (w, (Pj)
The other estimates are obtained similarly.

I

Definition 7.2. Let D(A) n (ker A)^ = ^ . Since A is a bijection of ^ onto R{A), let G be its inverse. G is a bijection of R{A) onto ^. We define G = G ( 1 + Q), and call G the Green operator of A. G is a linear map of L^(X, B) to ^ which coincides with G on R{A) and vanishes on ker A*. Theorem 7.4. T/ie Green operator G has the following properties: (r)

G /5 defined on L^(X, B), and its range R{G) is given by R{G) = W\X, B) n (ker A)^. For any u e L\X, B), AGu = {I- Q)u, and for any ve W\X,B), GAv = {I-P)v. (2°) For any integer /c ^ 0, there is a positive constant C^ such that for any u e W^(X, B), Gu e W^^\X, B), and ||Gw|U^,^Cfc||M|U

(7.12)

holds. If 0<6
(7.13)

450

Appendix. Elliptic Partial Differential Operators on a Manifold

(2°) For ueW\X,B), by Lemma 7.2, we have | | ( 1 - Q ) M | U ^ (Cfc + l)||w||fc. Let CJc be a positive constant in (7.6) of Lemma 7.L Then ||GW|U^, = ||G(1 + Q ) M | U ^ , ^ C U C , + 1)||M|U,

which proves (7.12). (7.13) is proved similarly.

I

Corollary. Let A(p, D) he a formally self-adjoint elliptic partial differential operator, P the orthogonal projection to ker A, and G the Green operator of Alfue C°°(X, B), then Gu e C°°(X, JB), and the following equality holds u = Pu + AGu. By this the orthogonal decomposition ^ ( X , B) = ker A e A C ° ' ( X , B) is given, where AC^iX, B) denotes the image of C^iX, B) by A{p, D). Proof Let A be the operator on the Hilbert space L^{X, B) defined from A{p, D) as in Definition 7.1. Since A{p, D) is formally self-adjoint, A is self-adjoint (Theorem 7.2). Therefore, since ker A = ker A*, the orthogonal projection P coincides with the orthogonal projection Q to ker A*. By Theorem 7.4, 1, for any u e C°°(X, B) c W\X, B), u = Pu + AGu holds. Furthermore by Theorem 7.4(2°), GueC°°{X,B), hence AGue AC^iX, B). Moreover since A is self-adjoint, by Theorem 7.3 (3°), ker A and AC^iX, B) are orthogonal to each other. I (b) The Spectrum of a Strongly Elliptic Partial Differential Operator Let A(p, D) be a strongly elliptic linear partial differential operator of order 2m. We define the operator A as in Definition 7.1. Let 8 be the constant of strong ellipticity of A(p, D), and take 5i, ^2 as in Theorem 4.1. If Re A > 5i, then for any (p e W^^(X, B), we have ||(A + A)(p||^(ReA-5i)||(^||.

(7.14)

For, indeed, by Theorem 4.2, we have ||(A + A)(^||||<^||^Re(A(p,(^) + R e A ( ( p , ( ^ ) ^ ( R e A - 5 i ) | k f . Furthermore there exists a positive constant C depending only on n, m, /A,

§7. Elliptic Operators in the Hilbert Space L^(X, B)

451

8, and M^ such that if Re A > 8„ for any (p e W^'^iX, B), \\cphm^C\\(A

+ X)cp\\

(7.15)

holds. In fact, it suffices to use A{p, D) +A for A(p, D) in (4.7), and make an estimate of ||<^||o by (7.14). Theorem 7.5. Let A{p, D) he a strongly elliptic partial differential operator. / / R e A > 5 i , thenA-^X is a linear isomorphism of W^"^(X, B) ontoL^(X,B). Proof. It is clear that A + A is a continuous injection of W^'"(X, JB) to L^(X, B). For any fe L^{X, B), there is a weak solution of the equation (A + A ) i / = / in W ( X , B) (Theorem 5.2). By Theorem 6.1, u e W^'^iX, B). Hence A + A is surjective. Therefore by (7.15) its inverse is also continuous. I Fix iji with iJi^Si, and put G^ = (A + M)-^

(7.16)

Since W^'^iX, B) c: L\X, B), G^ can be considered as a map of L\X, B) to V'{X,B). Then the following theorem follows from Rellich's theorem. Theorem 7.6. G^ is a compact liner map of L^{X, B) to L^{X, B).

I

For an arbitrary complex number A, consider the equation (A-A)
(7.17)

Multiplying by G^ from the left, and putting ^ = A + ^t, we have

(I-^G,)cp = GJ. Putting G^f=g,

we obtain {I-^G^)

(7.18)

We consider (7.18) instead of (7.17).

{A-\)

= {A-hfji){I-^GJ

= (I-CGJ{A

+ fi)

holds. Since (A + /x) is a bijection of \y^^(X, B) onto L\X, B), ( A - A ) is bijective if and only if I-^G^ is bijective. Thus we have the following theorem.

452

Appendix. Elliptic Partial Differential Operators on a Manifold

Theorem 7.7. A complex number A is contained in the spectrum of A if and only if ^~^ = (A — /JL)~^ is contained in the spectrum of G^. I Since G^ is a compact operator on L^(X, B), its spectrum is rather simple. In particular, except for 0, it has only the point spectrum whose only accumulation point is 0, and the generalized eigenspace belonging to each eigenvalue is finite dimensional. Theorem 7.8. Let A{p, D) he a strongly elliptic linear partial differential operator. Then the spectrum of A is contained in the half space Re A > — 5i, and it consists only of the point spectrum which has no finite accumulation point Furthermore the generalized eigenspace belonging to each eigen value is finite dimensional. I

§8. C^ Difierentiability of (p(0 In this section we shall prove that the vector (0, l)-form (p{t) constructed in Chapter 5, §3 is C°°. Let S be the disk of radius r in C " with the origin as its centre, namely, we put 5 = {^ = (^1,... ,^m)|kl^ = Z]li l^jl^^ ^^}- Let M be a compact complex manifold of dimension /i, and X its underlying C°° differentiable manifold. The vector (0, l)-form (p{t) on M parametrized hy teS satisfies the following quasi-linear partial differential equation.

(- 2 7-rT^+n)
x=\dt

dt

(s.i)

/

where [ , ] denotes the Poisson bracket. (p(t) is holomorphic in t, and C^^^ {k^2,l>0>0) with respect to p e X. Moreover we may assume that there exists a positive constant K such that \cp(t)\^^e^KAit),

(8.2)

A ( 0 = — I —ih + '-'^tJ^

(8.3)

where we put

1 6 c ^ = 1 /JL

with positive constants b, c. The aim of this section is to show that (p{t) is C°° on X x{r G C " 11 r| < 2"V} provided that r is a sufliciently small positive number. We cover M by coordinate neighbourhoods XjJ = \,... ,J. Let Zj = (zj,..., Zj) be the local complex coordinates on Xj with zf = xj' + v-Ixj"^". The vector-valued (0, l)-form (p(t) is represented on Xj in terms of these

§8. C°° Differentiability of
453

local coordinates as (p{t)=l
l cpUz,t)dz-d,.

A

(8.4)

X,a

Let 2ga0 dz" dz^ be the Hermitian metric tensor, and (g^") = (ga^)~^ Further we denote the covariant differentiation by V. Then if we put e[cp,cp]=l^Uz''d,,

(8.5)

we have cr,p,p

-"^.(pU.cp'^-cp^V^d^cp^^).

(8.6)

Since the problem is local, we can use the results from the theory of elliptic partial differential equations. Choose a C°° function (Oj(p) on X with supp (Oj c: Xj so that for any peX, i cOj(p)^l-

(8.7)

Next, for each / = 1, 2 , . . . , we choose a function rj\t) as follows. 77^(0^1

on

v\t)^0

on

| r | ^ ( 2 " ' + 2"'~')r, \t\^i2-'-^2-')r.

(8.8)

We assume that the r]^{t) are C°°. Put co'j(p,t) = coj(p)v'v(t)^

(8.9)

Furthermore, we choose a C°° function ;t'j(/^) with supp;^^c:A} which is identically equal to 1 on some neighbourhood of the support of coj. We put x'j{P,t) = Xj(p)v'it).

(8.10)

Since rj^it) is identically equal to 1 on a neighbourhood of the support of '^^^^(0, Xj is identically equal to 1 on some neighbourhood of the support of w]^\ First we shall prove that rj^ip is c^^^^^. (o^(p satisfies the equation

Appendix. Elliptic Partial Differential Operators on a Manifold

454

where we put Fo = Wje[
(8.12)

Here we use the fact that Xj is idetitically equal to 1 in a neighbourhood of the support of coj. The support of o)](p{p, t) is contained in Xj x S. In terms of local coordinates, we have <^j
= I o^'M t)
(8.13)

We introduce real coordinates x", a = 1, 2 , . . . , 2n + 2m, by putting /=!,...,n,

z^(;?) = x^ + V^x^'"",

By these 2n + 2m real coordinates, XjXS is identified with an open set Uj of a (2n + 2m)-dimensional torus j^^+^m A complex-valued function/(p, t) defined on X x 5 can be considered as a function on J^"^^'" if supp/c= Xj x 5. In this case we can consider the difference quotient A ^ / in the direction of x", a = 1 , . . . , 2n + 2m. Since the support of the vector-valued (0, l)-form co^cp is also contained in Xj xS, we define its difference quotient A^co]<;p as A'^co'jcpiz, t)=l

A'Aco'jCp',){z, t) dz' d„

(8.14)

A,p

Since (o](p^ is a complex-valued function whose support is contained in Xj X 5, the right-hand side is well defined. (8.11) can be considered as the equation on j^n+im HQ^icc^ by taking the difference quotient of both sides of this, we obtain the following equation. / \

^' + D]^Ala>]ct> y A^w](A ==F „F„ .=,3t^5f^

(8.15)

where we put

+ (nA^-A^n)cuj.p.

(8.16)

Since ( - Z r = i d^/St'' dt'' + D ) is an elliptic linear partial differential operator whose principal part is of diagonal type, by Theorem 2.3, we obtain the

§8. C°° Differentiability of (p{t)

455

following a priori estimate |A^a;j(p|,^, ^ C,(|FiU_2+, + |A^o>j^|o),

(8.17)

where Q is a positive constant, which may depend on k. In order to make an estimate of the right-hand side of this inequality, we use the following two lemmata. Lemma 8.1. Let u and v be complex-valued C^^^ functions defined on T' with k ^ 0 and 1> 6>0. Then the product uv is C^"^^, and there exists a positive constant Bj^ depending only on k and /, but independent ofu and v such that \uv\^^e^B^

I

(IwUall^ls + lt^Ui^U.).

(8.18)

r+s = k

The proof is easy, hence we omit it.

I

In order to make ah estimate of the norm of A«wJ(p, the following lemma is useful. This corresponds to Theorem 1.13. Lemma 8.2. Let Z j , . . . , x' be coordinate functions on the l-dimensional torus J^ and A^ the difference quotient operator in the direction ofx" by h. Then for any integer k^O and any 0 with 1> 6>0, we have the following: (i) (ii)

For Se C^"^^(T^, C'^), if\h\>0, the difference quotient A«5 is again an element of C'^'-'iT^C^. IfSe C^^^(J\ C^), then for any a = \,...J, and any h with 0 < |/j| < 1, the following estimate holds: |A^5k^a^ 151,^1-,,.

(iii)

(8.19)

IfSe C^^\t\ C^), andforanya = 1 , . . . , I and any h with Q<\h\ < 1, there exists a positive constant M such that |A^5|,^,^M,

(8.20)

thenSeC'''^'-'\j\C^). This lemma is an analogy of Theorem 1.13 with the Sobolev norm replaced by the Holder norm. The proof of Lemma 8.2 is, however, very easy unlike that of Theorem 1.13, hence we omit it. We shall make an estimate of || Fj || ^.20. The second term of the right-hand side of (8.12) is a linear combination of partial derivatives of order at most 1 of x]^' Also the third term of (8.16) does not contain the difference quotient of second-order partial derivatives of (o](p since they are cancelled.

456

Appendix. Elliptic Partial Differential Operators on a Manifold

Therefore, only the first term of F^ '2^'^co]e[cp,
(8.21)

contains the difference quotients of second-order partial derivatives of a)](p or Xj(p' Using the local representation (8.5) and (8.6), we can write (8.21) as

T,cr,p,v,\

-xW"^^. dAlicoh'i)}

X d r a, + • . •,

(8.22)

where we omit the terms involving no difference quotients of second-order partial derivatives of (o](p or x]^By Lemma 8.1, there exist positive numbers L, L' such that

\\^Wjeix]^.x]
Y. UiV;-|o|A^a),V,|,^, + L>,VU^el'A,VU...

(8.23)

By Lemma 8.1 and Lemma 8.2, the C^'^'^^ norm of the terms of Fj other than (8.22) can be estimated by a positive multiple of \o}](p\k+e\Xj]cp\j,^e + Mt,\co]cp\j,^e\x'j
(8.24)

where MQ and M^ are positive constants. By (8.17) and (8.24), we obtain

+ CkM,,\w](p\^^e\x](p\k^e'

(8.25)

We choose a sufficiently small r so that MoCfcA(r)<2-^

(8.26)

holds. Multiplying both sides of (8.25) by 2, transposing the first term of the right-hand side, and using (8.26), we obtain |A^cc>,VU+,^2CfcKVK + 2QMi|co,V|,+,|(A](pL^,.

(8.27)

§8. C°° Differentiability of
457

Since cp is C^^^, the right-hand side is bounded independently of h. This is true for any a = l,... ,2n + 2m. Therefore by (iii) of Lemma 8.2, (o](p is proved to be c^^^^^. Since this is true for any j = 1 , . . . , /, summing with respect to j , we see from (8.7) that r]^(p is a c^"^^^^ vector (0, l)-form. Next we shall prove that h^(p is c^^^"^^, using the same r determined by (8.26). Replacing co] by co] and Xj t)y x] in (8.15), and differentiating with respect to x^, we obtain (^-l^j^+nytD,w]cp = F„

(8.28)

where we put D^ = d/dx^, and F2 = DpF,-h{D^^^D^co^cp-DpD^^^(o]cp}.

(8.29)

By (8.28), we obtain the following a priori estimate: \A^^Dpw]
(8.30)

where Q is the same as in (8.17). |A^D^a;j(p|o is estimated by \(o](p\2. We want to obtain an estimate of 1^21^-2+0- Since the second term of the right-hand side of (8.29) does not contain the difference quotients of partial derivatives of order 3 of (o](p, the only term of F2 which involves difference quotients of the partial derivatives of order 3 of (o](p or x]

'2^aD,co]0[x]cp,x]
g'''{x]
T,cr,p,v,\ -XhT^cr

dAaiDp(0](p',)}

J r

a, + • ••,

(8.31)

where we omit the terms not involving difference quotients of partial derivatives of order 3 of a)](p or x]
+0X\(Oj
where M^+j may be different from M^, but MQ is the same as in (8.24). By (8.32) and (8.30), we obtain \^tDpCo]cp\^^e ^ CkMoA{r)\^'^DpCo]
458

Appendix. Elliptic Partial Differential Operators on a Manifold

Multiplying both sides of (8.33) by 2, transposing the first term of the right-hand side, and using (8.26), we obtain \^^Dpa)](p\j,+0 ^ 2CkMk+i\x^(p\k+i+e\o)](p\k+i+e -^2M^,^,\co]
(8.34)

Since rj^ip is c^^^'^^, the right-hand side is bounded independently of h. Since this is true for any a = 1 , . . . , 2n + 2m, by (iii) of Lemma 8.2, D^co](p is C^^^^^. Since this is true for any j8 = 1 , . . . , In + lm, we see that (o](p is C^^^^\ Summing with respect to j , we see from (8.7) that 17 V is C^""^"^^ vector (0, l)-form. Note that in the above we need not replace r satisfying (8.26) by a smaller one. Similarly we can prove that, for any / = 1, 2 , . . . , r]^^^^(p is C^"^^"^^, where we may choose r independent of /. Since rj^^^^{t) is identically equal to 1 on | r | < 2 " V which is independent of/, cp is C°° on X x l r e C ^ U r l <2"V}.

Bibliography

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[27] [28] [29] [30]

Index

Abelian variety 49 admissible fibre coordinate 325 affine algebraic manifold 40 algebraic curve 40 algebraic subset 40 algebraic surface 40 analytic continuation 9, 10 analytic function 10 analytic hypersurface 19 analytic subset 33 a priori estimate 275, 396, 430, 432, 436 arithmetic genus 220 associate 15 automorphism 43 base space 97, 184 Betti number 175 biholomorphic 26 biholomorphic map 26 biholomorphically equivalent 32 Bott's theorem 244 bracket 212 bundle canonical 164, 178 complex line 165, 166, 167 dual 102 tangent 96 trivial 99 canonical basis 358, 361 canonical bundle 164, 178 o f P " 178 Cauchy-Riemann equation 7 cell 169 Chern class 166 first 180 closed differential form 83 coboundary 115

cochain, ^-cochain 115 cocycle, ^-cocycle 116 codimension 234 cohomology 109 cohomology class 120 cohomology group 115 a-cohomology group 140 cf-cohomology group 139 with coefficients in a sheaf 116, 120 with coefficients in F 144 compact complex manifold 39 complete 228 complex analytic family 59 complete 228 effectively parametrized 215 induced by a holomorphic map 206 of hypersurfaces 234 trivial 61 complex dimension 29 complex Laplace-Beltrami operator 154 complex Lie group 48 complex line 29 complex line bundle 165, 166, 167 complex manifold 28, 29 compact 39 complex projective space 29 complex structure 28, 29, 38 complex torus 48 number of moduli of 238 complex vector bundle 101 component contravariant 153 CO variant 153 of a tangent vector 96 composite function 12 constant of ellipticity 392, 396, 431 constant of strong ellipticity 394, 399, 434 contravariant component 153 contravariant tensor field 107

462 contravariant vector 106 convergent power series 14 coordinate multi-interval 127 coordinate polydisk 32 coordinate transformation 29 coprime 15 covariant component 153 CO variant tensor field 106 covariant vector 106 covering manifold 45 covering transformation group 45 a-closed form 88 ^-closed vector (0, ^)-form 266 a-cohomology group 140 5-cohomology group with coefficients in F 144
Index Dolbeault's lemma Dolbeault's theorem dual bundle 102 dual form 150 dual space 102

134, 139, 140

effective 215 effectively parametrized 215 eigenfunction 323 eigenvalue 323 elliptic curve 47 complex analytic family of 68 elliptic modular function 69 elliptic operator with constant coefficients 391 elliptic partial differential equation 443 elliptic partial differential operator 363, 391, 430 strongly 320 elliptic surface 319 elliptic type 392, 395, 431 equicontinuous 276 equivalent 99, 109, 192 exact commutative diagram 133 exact differential form 83 exact sequence 123, 126 exterior differential 78

fibre 97 fibre coordinate 98 admissible 325 fine resolution 137 fine sheaf 134 fixed point 44 form a-closed 101 dual 150 harmonic 156 with coefficients in F 160 holomorphic p-form 88 with coefficients in F 143 1-form 77 (p, g)-form 86 with coefficients in F 143 r-form 77 (0, ^)-form with coefficients in F 142 formal adjoint 152, 439 formally self-adjoint 156, 321 Fourier series 373

463

Index

Fourier series expansion 368 Friedrichs' inequality 330 fundamental domain 47

Garding's inequality 399, 433, 435 geometric genus 220 germ 109 Green operator 275, 315, 323 group of automorphisms 43

harmonic differential form 144, 152, 156, 157 harmonic form 156 with coefficients in F 160 harmonic function 9 harmonic part 315 harmonic vector (0, <3f)-form 274 hermitian metric 144 on the fibres 158 Holder norm 274, 303, 403, 436 holomorphic 2 holomorphic function 1, 2, 30 holomorphic map 23, 30 holomorphic p-form 88 with coefficients in F 143 holomorphic vector bundle 101 holomorphic vector field 67, 69 holomorphically equivalent 32, 61 homogeneous coordinate, of P" 29 homomorphism, of sheaves 123 Hopf manifold 49, 55 Hopf surface 69 deformation of 208 hypersurface 40

inclusion 124 infinitesimal deformation 182,188,190 along d/dt 198 infinitesimal displacement 236 inner product 102, 147, 425 integrability condition 269 integral domain 15 intersection number 54 irreducible 15 irreducible equation 20 irreducible factor 15 irreducible factorization 15 irregularity 220 isomorphism, of sheaves 124

Jacobian 24 Jacobian matrix

24

Kahler form 174 Kahler manifold 174 Kahler metric 174 kernel 123 Kronecker product 104 Kuranishi family 318 Laplacian 9 Lax-Milgram theorem 440 Leibnitz' formula 367 length, of a multi-index 363 line at infinity 57 linear partial differential operator local C°° function 109 local C°° vector (0, ^)-form 256 local complex coordinate 28 local coordinate 29 with centre q 32 local equation 33 local homeomorphism 110 locally constant function 112 locally finite open covering 32 locally trivial 193 majorant 277 manifold affine algebraic 40 compact complex 39 complex 28, 29 differentiable 37, 38 projective algebraic 40 topological 37, 38 map biholomorphic 26 holomorphic 23, 30 meromorphic function 35 minimal equation 22 mixed tensor field 107 modification 53 monoidal transformation 319 multiple 15 multiplication operator 423 Noether's formula 220 non-singular prime divisor 167 non-homogeneous coordinate, o f P " 30

424

464 norm 148 number of moduli 314

Index

215, 227, 228, 305,

obstruction 209, 255 first 214 Kh 255 1-form 77 orbit 43 order 321 orientable 80 orientation 80, 86

parameter 184 parameter space 60, 184 parametrix 405 partial derivative, of a distribution 366 period matrix 48 Poincare's lemma 83 point at infinity 57 polydisk 2 power series ring 14, 15 (p, ^)-form 86 with coefficients in F 143 principal part 154, 395, 431 principal symbol 392, 395, 431 projection 97, 112 projective algebraic manifold 40 projective space 174, 216 complex 29 homogeneous coordinate of 29 non-homogeneous coordinate of properly discontinuous 44

^-cochain 115 ^-cocycle 115 quadratic transformation quartic surface 307 quotient sheaf 125 quotient space 43

r-form 77 rank 97 rational function 42 real vector bundle 101 reducible 15 refinement 117 region of convergence 7

57

30

regular 17 relatively prime 15, 19, 35 Rellich's theorem 427 restriction 114, 171, 172 Riemann matrix 49 Riemann-Roch formula 227 Riemann-Roch-Hirzebruch theorem 220, 221 Riemannian metric 321

sheaf 109, 112 fine 134 of germs of C°° functions 111 of germs of C°° sections of F 112 of germs of C°° (0, ^)-forms with coefficients in F 142 of germs of holomorphic functions 111 of germs of locally constant functions 112 quotient 125 simplicial decomposition 168 singular point 33 smooth 33 Sobolev norm 372, 430 Sobolev space 376, 425 Sobolev's imbedding theorem 369, 370, 428 Sobolev's inequality 330, 428 space of moduli 233 spectrum, of a strongly elliptic operator 450 stalk 112 strongly elliptic 156, 322, 394, 399, 434 strongly elliptic partial differential operator 320 structure theorem of distributions 375 subbundle 107 submanifold 33, 34, 39 subsheaf 123 support 82, 134, 367 surgery 52, 53 system of local complex coordinates 28, 29 system of local coordinates, of a differentiable family 185 system of local C°° coordinates 37, 38

tangent bundle 96 tangent space 94, 96

465

Index

tangent vector 62, 95, 96 tensor field 105 contravariant 107 covariant 106 mixed 107 tensor product 103, 104 theorem of completeness 230, 284 topological manifold 37, 38 transition function 98 translation 367 trivial 61, 99, 193

holomorphic 101 rank of 97 real 101 vector field 63 holomorphic 67, 69 vector-valued distribution vector (0, ^)-form 5-closed 266 harmonic 274 local C ^ 256 volume element 146

unique factorization domain 15 unit 15 upper semicontinuous 200, 326, 351, 352

weak solution 438, 439 Weierstrass P-function 46 Weierstrass preparation theorem Whitney sum 108

vector bundle 94, 97 complex 101

(0, q)-foTm with coefficients in F

368

13

142