Complex Manifolds (AMS Chelsea Publishing)

COMPLEX MANIFOLDS JAMES MORROW KUNIHIKO I(oDAIRA AMS CHELSEA PUBLISHING American Mathematical Society· Providence, Rho...

Author: James Morrow and Kunihiko Kodaira

100 downloads 1298 Views 8MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

COMPLEX MANIFOLDS JAMES MORROW KUNIHIKO I(oDAIRA

AMS CHELSEA PUBLISHING American Mathematical Society· Providence, Rhode Island

2000 Mathematics Subject Classification. Primary 32Qxx.

Library of Congress Cataloging-in-Publication Data Morrow, James A., 1941Complex manifolds / James Morrow, Kunihiko Kodaira. p. cm. Originally published: New York: Holt, Rinehart and Winston, 1971. Includes bibliographical references and index. ISBN 0-8218-4055-X (alk. paper) 1. Complex manifolds. I. Kodaira, Kunihiko, 1915- II. Title.

QA331.M82 2005 515'.946---dc22

20051

© 1971 held by the American Mathematical Society. Reprinted with errata by the American Mathematical Society, 2006 Printed in the United States of America. §

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321

11 10 09 08 07 06

Preface

The study of algebraic curves and surfaces is very classical. Included among the principal investigators are Riemann, Picard, Lefschetz, Enriques, Severi, and Zariski. Beginning in the late 1940s, the study of abstract (not necessarily algebraic) complex manifolds began to interest many mathematicians. The restricted class of Kahler manifolds called Hodge manifolds turned out to be algebraic. The proof of this fact is sometimes called the Kodaira embedding theorem, and its proof relies on the use of the vanishing theorems for certain cohomology groups on Kahler manifolds with positive lines fundles proved somewhat earlier by Kodaira. This theorem is analogous to the theorem of Riemann that a compact Riemann surface is algebraic. This book is a revision and organization of a set of notes taken from the lectures of Kodaira at Stanford University in 1965-1966. One of the main points was to give the original proof of the Kodaira embedding theorem. There is a generalization of this theorem by Grauert. Its proof is not included here. Beginning in the mid-1950s Kodaira and Spencer began the study of deformations of complex manifolds. A great deal of this book is devoted to the study of deformations. Included are the semicontinuity theorems and the local completeness theorem of Kuranishi. There has also been a great deal accomplished on the classification of complex surfaces (complex dimension 2). That material is not included here. The outline is roughly as follows. Chapter I includes some of the basic ideas such as surgery, quadric transformations, infinitesimal deformations, deformations. In Chapter 2, sheaf cohomology is defined and some of the completeness theorems are proved by power series methods. The de Rham and Dolbeault theorems are also proved. In Chapter 3 Kahler manifolds are studied and the vanishing and embedding theorems are proved. In Chapter 4 the theory of elliptic partial differential equations is used to study the semi-continuity theorems and Kuranishi's theorem. It will help the reader if he knows some algebraic topology. Some results from elliptic partial differential equations are used for which complete references are given. The sheaf theory is self-contained. We wish to thank the publisher for patience shown to the authors and Nancy Monroe for her excellent typing. James A. Morrow Kunihiko Kodaira

Seattle, Washington January 1971 v

Contents Preface

v

Chapter 1. Definitions and Examples of Complex Manifolds 1. Holomorphic Functions 2. Complex Manifolds and Pseudogroup Structures 3. Some Examples of Construction (or Description) of Compact Complex Manifolds 4. Analytic Families; Deformations

1 1 7 11 18

Chapter 2. Sheaves and Cohomology 1. Germs of Functions 2. Cohomology Groups 3. Infinitesimal Deformations 4. Exact Sequences 5. Vector Bundles 6. A Theorem of Dolbeault (A fine resolution of (I))

27 27 30 35 56 62 73

Chapter 3. Geometry of Complex Maoifolds 1. Hermitian Metrics; Kahler Structures 2. Norms and Dual Forms 3. Norms for Holomorphic Vector Bundles 4. Applications of Results on Elliptic Operators 5. Covariant Differentiation on Kahler Manifolds 6. Curvatures on Kahler Manifolds 7. Vanishing Theorems 8. Hodge Manifolds Chapter 4. Applications of Elliptic Partial Differential Equations to Deformations 1. Infinitesimal Deformations 2. An Existence Theorem for Deformations I. (No Obstructions) 3. An Existence Theorem for Deformations II. (Kuranishi's Theorem) 4. Stability Theorem

83 83 92 100 102 106 116 125 134

147 147 155 165 173 186 189 193

Bibliography Index Errata

vii

Complex Manifolds

[1]

Definitions and Examples of Complex Manifolds I.

Holomorphic Functions

The facts of this section must be well known to the reader. We review them briefly. DEFINITION 1.1. A complex-valued function J(z) defined on a connected open domain W s;;; en is called hoiomorphic, if for each a = (a1> "', an) e W, J(z) can be represented as a convergent power series +00

L

ek, ... kn (Z1

a 1)k, ... (zn - a,,)k"

-

k,~O.kn~O

in some neighborhood of a. REMARK. If p(z) = LCk ... k n (Z1 - a1)k, •.• (z" - an)k" converges at z = w, then p(z) converges for any z such that IZk - akl < IWk - akl for 1 :S k :S n.

Proof We may assume a = O. Then there is a constant C> 0 such that for all coefficients Ck .... kn '

Iek, .. ·kn W"l1

.••

wknl , ,< _C .

Hence

I

Iek, ... k zk,1 ... zknl ,,< - C 2Z Ik' '" n

W1

I

2Z

Ik" •

(1)

W"

If Izdwil < 1 for 1 :S i:S n, (1) gives

L Ie", "'knZ~'

'"

zktl

n( 1

:S C.1=1

Zi

1- -

Wi

1

I)

< + 00.

Q.E.D.

2

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS We have the following picture:

Figure I

n is the region {zllzil < Iwd i ~ n}. For convenience, we let P(a,r) = {zllz. - a.1

< r., v = 1, "', n}.

Sometimes we call Pea, r) a po/ydisc or po/ycylinder. A complex-valued function/(z) = /(x1 + iYI, ... , Xn + iYn), where i = 1 can be considered as a function of 2n real variables. Then:

J-

DEFINITION 1.2. A complex-valued function of n complex variables is continuous or differentiable if it is continuous or differentiable when considered as a function of 2n real variables. We have: THEOREM 1.1. (Osgood) If fez) = /(Zl' " ' , Zn) is a continuous function on a domain W £ en, and if / is holomorphic with respect to each z" when the other variables Zi are fixed, then/is holomorphic in W. Proof Take any a E Wand choose r so that pea, r) ~ W. We use the Cauchy integral theorem for the representation for Z E Pea, r) f( ZI, . . . , f( WI>

and so on.

Z2'

...

) - _1 f Z" -

J,

• 2Xl Iw,-lId=r,

)- 1 2x!

,Z" - - .

f.

Iwz-lIzl=rz

f(w l ,

z2, ... ,

WI -

Zn) d wI>

Zl

f(w l ,W2 ,Z3,···,z")d

W2,

W2 -

Z2

1.

HOLOMORPHIC FUNCTIONS

3

Substituting we get

We are assuming

< 1. Iz.w. -- a'l a. Hence the series

1 w. -

Z.

1

= (w. -

=(

+ (a. - z.) =

a.) 1

)

w. - a.

[ 1-

(Zy -

1 ] 1 ay/w. - aJ w. - a.

L (Z - a )k 00

v

k=O

v

w. - a.

converges absolutely in P(a, r). Integrating term by term we get 00

J(z) =

L ct

! •••

kn(zt - a 1 )k!

•••

(zn - an)kn,

(2)

n=O

where

Then

where M = sup{IJ(w)llw E P(a, r)}. It follows that the representation (2) for J(z) is valid for Z E P(a, r) and hence the theorem is true. We now introduce the Cauchy-Riemann equations. Let/(z) be a differentiable function on domain n f; en. DEFINITION

a/az a/oz., 1 ~ v ~ n are defined by af 1 (aJ . Of) o~. = 2 ax. - I oY. ' af 1 (a f . OJ) oz. = 2 OX + I Oy. '

1.3. The operators

y ,

y

where z. =

Xy

+ iy. as usual.

4

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS Let f(z) = u(x, y)

+ h'(x, y). Then

of = ~ [au + i av + i(au + i av)] az 2 ox ax ay oy =

°

~ [OU _ ov + i(OV + aU)]. 2 ox oy ox oy

So, af/oz = if and only if ou/ax = oll/ay and or/ax = -ou/ay (the CauchyRiemann equations). REMARK. If of/oz = 0, then df/dx = of/oz, where df/dx = ou/ox + i(ov/ox). The following calculation verifies this:

of = ~ [au + i ov _ i(OU + i Ov)] 2 ax ax ay ay

OZ

=

~ [OU + i OV + i (av _ i au)] . 2 ax ax ax ax

THEOREM 1.2. Let fez) be a (continuously) differentiable function on the open set Q s;;; en. Thenf(z) is holomorphic if and only if of/oz. = 0, i :s v :S n.

Proof This follows easily from Osgood's theorem and the classical fact for functions of one complex variable. We need another simple calculation. From now on differentiable will mean having continuous derivatives of all orders (C""). PROPOSITION 1.1. Suppose few) =f(w1, ... , wm ) and 9 ..(Z) I:s A.:s mare differentiable and such that the domain off contains the range of (91' ... , 9 ..) = 9. Then f[91(Z), .. " 9m(Z)] is differentiable and if w;.(z) = 9;.(z),

of = oz.

f

(Of ow;. + !L ow;.) oz. ow). oz. '

..= lOW).

(3)

(4) Proof All statements follow trivially from the chain rule of calculus. For punishment we calculate (3). Let 11'). = U). + iv). = 9.(z). Then

I.

5

HOLOMORPHIC FUNCTIONS

Making the substitutions, 1

UA

= 2(g A + 9 A),

we get

f

oj[g(Z)] = {OJ! (09A + 09A) OZ, A= I oU A2 oz. oz. oj

+ OVA =

(1 )(09A 09A)} OZ. - OZ. 2i

f {2I (OJOUA -

A= I

. I

Of) og A OVA oz.

. of) 09 +-I (oj -+1- A} 2 OUA

OVA OZ.'

which gives (2). COROLLARY 1. If f(w) is holomorphic in wand if w = g(z) = [gl(z), "', gm(z)] where each g;.(z) is holomorphic in z, thenf[g(z)] is holomorphic in z.

COROLLARY

2.

The set ()n of all functions holomorphic on

n forms a ring.

In order to study complex manifolds we must consider holomorphic maps. Let U be a domain in en and letfbe a map from U into em, f(Zl' '.', zn) = [ft(z), ... ,fm(z)]. DEFINITION

1.4. f is holomorphic if each f;. is holomorphic. The matrix

ojl OZI

ojm OZI

ojl OZn

ojm OZn

=

(iz:);.=I . . .

m

v= 1, ...• n

is called the Jacobian matrix. If m = n, the determinant, det(of;./ozv) is called the Jacobian. Writing out the real and imaginary parts W;. = U;. + iv;. = f;., z. = x. + iy., we have 2n functions U;., V;. of 2n real variables x., y •. We write briefly

6 REMARK.

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

IfI is holomorphic, a(u, v)/a(x, y) = Idet(al.,jaz.)I Z ~

o.

Proof We write it out for n = 2 and leave the general case to the reader. We use the Cauchy-Riemann equations and set a. A= aUA/aX. = aVA/ay., bVA = aVA/aX. = -au}../ay•. Then

au, ax,

av, ax,

oU z av z ax, ax,

aUI ay,

av,

aU 2 av z = ay, aYI

aYI

all

b ll

al2

bl2

-b'l

all

-bJ2

al 2

a21

b21

a22

b 22

-b 21

a2'

-b 22

a22

We perform the following sequence of operations: Multiply column 2 by i and add it to column I ; do the same with columns 4 and 3. Then multiply row 1 by i and subtract it from row 2; do the same with rows 3 and 4. Making use of the fact that B.A = aIA/aZ. = a. A+ ib. A, we get gil a(u, v) gZI --= 0 o(x, y) 0

gl2 g22 0 0

* *

* *

gz,

gZ2

gil gl2

= Idet(g.A)1 2

by interchanging columns 2 and 3 and rows 2 and 3.

Q.E.D.

THEOREM 1.3. (Inverse Mapping Theorem) Let/: V -+ en be a holomorphic map. If det(oJ,./oz.)lz= .. :F- 0, then for a sufficiently small neighborhood N of a,Jis a bijective map N -+ I(N);J(N) is open and/-'I/(N) is holomorphic on/(N). Proof The remark gives o(u, L,)/a(X, y) :F- 0 at a. We then use the inverse mapping theorem for differentiable (real variable) functions to conclude that I(N) is open, I is bijective, and I-I is differentiable on I(N). Set qJ(w) = /-I(W); then z" = cp,,[J(z)]. Computing,

o = a~1l = az.

±

aCPIl a~A A=I aw}.az.

+ a~" a~A

awAaz.

But det(alA/az.) = det(a/A/az.) :F- O. So by linear algebra, aqJ,,/aWA = 0 and qJ =/-1 is holomorphic. Q.E.D.

2.

7

COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES

COROLLARY. (Implicit Mapping Theorem) Letf)., A. = I, ... , m be holomorphic on V ~ en. Let rank (fJf)./fJz.) = r at each point z of V and suppose in fact that det(iJf;./fJzvhsr :# o. Iff;.(a) = 0 for A S; m for some a E V, then in vsr

a small neighborhood of a, the simultaneous equations,

have unique holomorphic solutions AS; r.

F or more details in this section one may consult Dieudonne (1960).

2.

Complex Manifolds and Pseudogroup Structures

We assume given a paracompact Hausdorff space X which will also generally be assumed connected. We want to define what we mean by a complex structure on X (or structure of a complex manifold) which will be an obvious generalization of the concept of a Riemann surface. First we want to assume X is locally homeomorphic to a piece of C". DEFINITION 2.1. By a local complex coordinate on X we mean a topological homeomorphism z:p -+ z(p) E C" ofa domain U ~ X. z(p) = [Zl(p), ... , z"(p)] are the local coordinates of X. DEFINITION 2.2. By a system of local complex analytic coordinates on X we mean a collection {Zj}jEI (for some index set I) of local complex coordinates Zj: Vj -+ C" such that: (I)

X=UU J • JEI

(2) The maps fjk: Zk(P) -+ Zj(p) are biholomorphic [that is, Zj Zk- 1 = fjk and r;,/ = Zk zj I are holomorphic maps from Zk( Vj n Vk) onto Zj(Vj n Vk)] for each pair of indices (j, k) with Vj n V k :# ljJ. 0

0

DEfiNITION 2.3.

Two systems

{Zj}jd' {II').}).'A

are equivalent if the maps

Zj(p) -+ w).(p) are biholomorphic when and where defined.

DEfiNITION 2.4. By a complex structure on X we mean an equivalence class of systems of local complex (analytic) coordinates on X. Bya complex manifold M we mean a paracompact Hausdorff space X together with a complex structure defined on X.

8

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

EXAMPLE, Complex projective space lPn, This is constructed from {O} by identifying (p '" q)p = (pO, pI, ... , pn) and q = (qO, ... , qn) if and only if pA = cqA for some nonzero c E C, for 0 ~ A. ~ n. Then IP n = en + I {O}/'" is a compact Hausdorff space and one can construct a system of complex coordinates as follows: We let Vj = {p E IPnlpj ¥- O}. Then {Vj}jsn is an , f 10. rrM 0 n V j themapzj= (ZO n) h opencovenngo j , " 'J ,Z j I ,Zjj+ I ,,,·,zj,were z/ = pA/pj gives a local coordinate on V j ; in fact, Zj(V) = en. Then fjk: Zk --+ Zj is given by zj = z:/zt for A =F k, z~ = I/z{. (One simply multiplies by pk/p j ,) Thus we see that {V j , zJ is a complex analytic system defining a complex structure on IPn.

en+ 1 _

Generalizing this procedure we introduce the idea of a pseudogroup structure. All spaces will be Hausdorff in what follows. 2.5. A local homeomorphism f between two spaces X and Y is a homeomorphism of an open set V in X to an open setf(V) in Y. One has a similar definition of local diffeomorphism. A local homeomorphism (diffeomorphism) of X is such a map with X = Y. Let 9 be a domain of IR n or en. Letfand 9 be local diffeomorphisms of 9. If open W £:; 9, fl W denotes f restricted to W which is the restriction off to domain (f) n W. If W is some open set such that 9 is defined on Wand W nf(V) ¥- 4l. then 9 of is defined onf-I[W nf(V)], DEFINITION

feU)

Figure 2

DEFINITION 2.6. A pseudogroup of transformations in 9 is a set r of local diffeomorphisms of 8 such that

(I) fEf=:.I- I Er. (2) fE r, 9 E r = go IE r where defined. (3) fE r=/1 WE r for any open W£:; 8.

(4) The identity map id E r. (5) (completeness) Let I be any local diffeomorphism of 9. If [} = u Vj andll Vj E r for eachj, thenfE r.

2.

COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES

9

2.7. Let r (a pseudogroup on 9) and X (a paracompact Hausdorff space) be given. By a system of local r-coordinates we mean a set {ZjLd of local topological homeomorphisms Zj of X into 9 such that Zj a Z;;l E r whenever it is defined. {w;.} and {Zj} are equivalent (f-equivalent) if W;. a zj' E r when defined. A r-structure on X is an equivalence class of systems of local r-coordinates on X. A r-manifold is a paracompact Hausdorff space X together with a r-structure on X. DEFINITION

EXAMPLES 1. 9 = en, re = (all local biholomorphic maps of e").Thenarc-structure is a complex structure, and a re-manifold is a complex manifold. 2. 9 = ~", fd = (all local diffeomorphisms of ~n). Then a fd-structure is a differentiable structure and a fd-manifold is a differentiable manifold. 3. Let r be the set of a local diffeomorphism / of ~2" satisfying the following condition. The matrix (e;..) will be defined to be

0

-1

1

0

0

-1

0

0 0

0

-1

1

0

(? -

where the blocks ~) occur on the diagonal and the rest of the entries are zeros. If x = (x', ... , x2n) E ~2n,f(x) = [!t(x), ... ./2ix)] then the derivatives of / should satisfy

A system satisfying Example I is called a Hamiltonian dynamical system, and such an / is a canonical trans/ormation. In this case a f-structure is called a canonical structure. 4. Let r = (local affine transformations of ~"). These transformations have the form n

/A(X) =

L a! x + b\ Y

Y= ,

where the a~, b;' are constants and the matrix (a~) is nonsingular. In this case a f -structure is called flat affine structure. If pseudogroup structures f, and f 2 are such that f, c f 2' then every system of local f, coordinates is a system of local f2 coordinates, and f, equivalence implies r 2 equivalence. Hence, every f,-structure determines a

10

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

f 2 -structure. By assumption f c fd for all f. So every f-structure on X determines a differentiable structure on X and every f-manifold is a differentiable structure on X and every f-manifold is a differentiable manifold. The f-structure M is defined on the differentiable manifold X. The problem of determining the f-structures on a given differentiable manifold M for given f is one of the most important (and difficult) problems in geometry. It is known, for example, that if X is a compact orientable differentiable surface (real dimension 2), then the only complex structures on X are those of the classical Riemann surfaces. I n case X = S2 (as a differentiable manifold), then X = pI complex analytically (this is a classical fact). If the underlying differentiable manifold X is diffeomorphic to pn, then one conjectures that X = pn complex analytically [see Hirzebruch and Kodaira (1957)J, and Kodaira and Spencer (1958). If S211 is the sphere with its usual differentiable structure, it can be shown [Borel and Serre (1953) and Wu (1952)] that s2n for n =/; 1,3 has no complex structure 2n + I

[s2n

=

{(Xl' ••• , X2n+l) 1

i~2

xf, (Xl'···' X 2n + I ) E 1R2n+I}J.

For S2 there is the usual complex structure. It has been recently proved by A. Adler (1969) that S6 has no complex structure. As a final example, let M be a compact surface and let f+ be the pseudogroup of all local affine transformations, v = 1,2 such that

We have: THEOREM 2.1. [Benzecri (1959)] If a f+ -structure exists on M, then the genus of M is I. If M is not a torus, then M cannot be covered by any system {(x), X])} of local coordinates such that lax~/ax;;1 is constant on Uj n Uk for each pair of indices (j, k). The proof will not be given here. We continue with the definitions. Let M be a complex manifold, Wan open set in M, and {Zj} a coordinate system. Then a mapping I: W ~ C l is holomorphic (difJerentiable, and so on) if I zj I is holomorphic (d(fJerentiable, and so on) for eachj where defined. Let N be another complex manifold with coordinates {II";.} and I: W -. N. Then I is holomorphic (differentiable, and so on) if lI"A 0 I 0 zj I is holomorphic where defined. 0

DEFINITION 2.8. A subset SsM of a complex manifold is a (complex) analytic subvariety if, for each S E S, there are holomorphic functions IA(P) defined on a neighborhood lJ 3 S, 1 :::;; A. :::;; r, such that S n U = {p I/ip) = 0,

3.

CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS

11

I ~ A. ~ r}. Then fA = 0, I ~ A. ~ n, are the local equations defining S at s.

The subvariety S is called a submanifold if S is defined at each s E S by local equationsf. = 0 such that iJf;.(P)] ran k [- = r. IS .10 d epen d ent 0 f s. azj(p) Suppose det(afA/azj)1 ~A:Sr =F O. Then letting Isv,;,

w7(p) =

lip),

w;(p) = z;(p),

for A = I, "', r for A = r

+ 1, ... , n,

we have a local coordinate li'i = (wJ, "', wi» such that S: wJ = 11'] = ... = wj = 0 (is defined by). Let (;(p) = IV'/A(p) = zj+A(p) for PES n V j • Then S is a complex manifold with local coordinates gj}' We want to introduce meromorphic functions on a complex manifold. They should be those functions which are locally quotients of holomorphic functions. More precisely: 2.9. A meromorphic function f on M is a complex-valued function defined outside of some proper subvariety S of M (S =F M) and such that given q EM, there is a neighborhood V of q in M and local holomorphic functions g, h on V such thatf(p) = g(p)/h(p) for p E V - S. DEFINITION

EXAMPLES l. Any holomorphic mapf:M ..... [pI1 = C U {co}, [S =f-I(oo)]. 2. In C 2,j(Zl> Z2) = ZI/ZZ or f(zl' Z2) = P(ZI' Z2)/Q(Zl' Z2)' where P and Q are polynomials.

3. Some Examples of Construction (or Description) of Compact Complex Manifolds First we have submanifolds of known manifolds ([pi", [pili X [pi", and so on). Let [pi" have homogeneous coordinates «(0' "', (,,). Let fi 0, I ~ A ~ m be homogeneous polynomials and define M = {( IfiO = 0, I ~ A ~ m}. Suchan M is called a projective algebraic (or simply algebraic) variety. If the rank of (afA/a(.>c is independent of (E M, then M is a complex manifold. These are exactly the classical algebraic (projective algebraic) manifolds. In some cases the equationsfA = 0 give some easily read information about M. For instance, if f is homogeneous of degree d, then Md = {Clfm = O} is called a hypersurface in [pi" of order d. If at least one of (ofliJ().)«) =F 0, I ~ A. ~ n, for each ( E M d, then M d is nonsingular.

12

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

EXAMPLES 1. Md S;; 1FD2 a nonsingular plane curve of order d is a Riemann surface of genus 9 = td(d - 3) + I. 2. A nonsingular M d £; 1FD3. M d is simply connected and the Euler number X(M d) = d(d 2 - 4d + 6). [The formulas in Examples 1 and 2 can be obtained from Hirzebruch (1962), p. 91, Equation (5). They are well-known classical formulas. The simple connectivity is also well known and it follows from the Lefschetz theorems on hyperplane sections-see Milnor (I963), p. 41.] 3. Let M £; 1FD3 be defined by

M = {(\(.(2 - (0(3 = 0, (0(2 - (~= 0, (~- (1(3 = O}. We claim that M is complex analytically homeomorphic to pl. One can easily check that the map fJ: IFDI --.IFD 3 defined by fJ(t) = (t5, t~tl' tot~, tn where t = (to , t,) E IFD I, is a biholomorphic map of IFD I onto M. We remark that in the cases of complex or differentiable structures, submanifolds give many examples; but for general i-structures one does not usually get sub i-structures. Second we get quotient spaces. DEFINITION 3.1. An analytic automorphism of M is a biholomorphic map of M onto M. The set of all analytic automorphisms of M forms a group 9 with respect to composition. Let G £; 9 be a subgroup. DEFINITION 3.2. G is called a properly discontinuous group of analytic automorphisms of M if for any pair of compact subsets K" K2 £; M, the set {g E G I gK, n K2 =t=
G has no fixed points if for all 9

E

G, 9 =t= 1, 9 has no fixed

THEOREM 3.1. If G is properly discontinuous and has no fixed point, then the quotient space MIG is a complex manifold in an obvious natural manner. Proof We shall assume that M is connected (or a countable union of connected manifolds) and paracompact. Hence, M is u-compact (a countable union of compact sets). Let MIG = {G p \ P EM}, where Gp = {g(p) \ pEG} are the orbits of p E M. As notation set MIG = M*, Gp = p*. We shall show that given q E M we can choose a neighborhood V of q such that PI' P2 E V, PI =t= P2 gives P; =t= p;. In fact, there is V 3 q such that gV ("\ V =

V 2 => V3 ... be a base of rela-

3.

13

CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS

tively compact neighborhoods at q. Then Fm = {g IgVm n Vm =F cp} is a finite subset of G and Fm;;2 Fm +S' ;;2 •••• If 3gmE Fm, gm =F 1 for all m, then since each Fm is finite, n Fm 3 g, 9 =F 1. Therefore, gVm n Vm =F cp, for all m and Vm ~ q, gives g(q) = q, contradicting the nonexistence of fixed points. Hence we cover M with open sets Vj such that PI' P2 E Vj implies pi =F p~ and thus, Vj ~ V; = {p* I P E Vj} is I - 1. We give V; the complex structure that Vj has. That is, if Zj: P ~ Zj(p) is a local coordinate on Vj , then zj: P* ~ zj(p*) = Zj(P) gives a local coordinate on M*. The system {zj} then defines a complex structure on M* and the topology of M* is just the quotient topology for the map M ~ M*. Q.E.D. EXAMPLES Complex tori. Let M = cn. Take 2n vectors {WI' . " , w2n }, (Wkl' " . , w kn ) E Cn so that the Wj are linearly independent over lit Let

1.

Wk

=

2n

G={glg:z~g(z)=z+ Lmkwk,mkEZ}. k=1

Tn = en/G isa (complex) torus of complex dimension n. Let n = I and arrange it so that WI = I, W z = w, where the imaginary part of W is positive. Then T=CI/G.

Figure 3 exp 2,,/

2 I

We have a map C C*, z~ w = e "z where C* = {zlz =F O}.lfwe first take g(z) = z + mlw + m2 and then exponentiate, we get e2Iti(z+mlwl. So exp 21ti 0 9 = oe ml • exp 21ti where oe = e2"iw and g(z) = z + mlw + m2' and 0< lexl < 1 since Im(w) > O. Looking a little closer we see we have the diagram

C~C*

·1 I~' n".'

C-C

which commutes. Hence, if we let G* = {g* I g*: w ~ exmw, me Z}, we see T = C/G = C*/G*.

14

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

Figure 4

= eN - {O} and G = {gIn I m E oZ, g(Wl' ... , where < l(Xvl < I}. Then WIG is a compact complex manifold since it is easy to see that G is properly discontinuous and has no fixed points on W. It is also easy to see that WIG is diffeomorphic to 2.

wN )

Hop! manifolds. Let W

= «Xl 11'1' ••• , (XN wN ),

Sl x

°

S2N-l.

3.

Let M be the algebraic surface (complex dimension 2) defined:

M = {( I

a + (i + (i + ,~ = o} £

p3.

Let G = {gm 1m = 0, 1,2,3,4 where g«(o, ... , (3) = (p(o, p2( 1, p3(2' p4(3) and p

= e21ti / 5}.

Then 9 is a biholomorphic map p3 ~ p3 and g5 = 1. Consider the fixed points of gm on p3. They satisfy (0 = v ~ 3), (p,"(v+l) - c) (. = and the fixed points are (l, 0,0,0), (0, I, 0, 0), (0,0, I, 0), and (0, 0, 0, I). These points are not on M so there are no fixed points on M and M /G is a complex mamfold. We saw before that M is simply connected and X(M) = d(d 2 4d + 6) where d = 5. Therefore, the Euler number of M is 55. Then the fundamental group 1C 1(M/G) ~ G and x(MIG) = II. 4. Last we have the classical examples of Riemann surfaces and their universal covering surfaces. If S is a compact Riemann surface of genus 9 ~ 2, the universal covering surface of S is the unit disk D = {z E e11lzl < I}. Then S = D/G where each element of G is an automorphism of D and hence of the form

°

g(z)

. z - (X (Xz - I

= el8 - - ,

I(XI < l.

3.

CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS

15

Finally we consider surgeries. Given a complex manifold M and a compact submanifold (subvariety) ScM, suppose we also have a neighborhood W => S and manifolds S* c W* with W* a neighborhood of S*. Suppose j: W* - S* -.... W - S is a biholomorphic map onto W - S. Then we can replace W by W* and obtain a new manifold M* = (M - W) u W*. More precisely, M* = (M - S) u W* where each point z* e W* - S* is identified with z = j(z*).

-

f

[-~J

Figure 5

Hirzebruch (1951) Let M = 1Jl>' X 1Jl>'. In homogeneous coordinates, 1Jl>' = {C/ ( = «(0' ~,)}; = {C u {(Xl in inhomogeneous coordinates, (= (d(o e C u too}. M = 1Jl>' X 1Jl>' = {(z, 0 I z e 1Jl>', (e 1Jl>1} contains S = to} X IJl>I and W = D X 1Jl>' where D = {zllzl < e} is a neighborhood of Sin M. Let W* = D X 1Jl>'* = {(z, (*) I zeD, (* e 1Jl>'*} and S* = to} x 1Jl>*. Fix an integer m > 0 and define j: W* - S* -.... W - S as follows: EXAMPLE I.

j(z, (*) -.... (z,

0

= [z,«(*/z'")]

where 0 < Izl <

B.

Then j is biholomorphic on W* - S* and let M! = (M - S) u W* where 0 = (z, (*) if (* = zln(, 0 < Izl < f:.

(z,

REMARK.

M and M! are topologically different if m is odd.

(for m = I). M = 1Jl>' X IJl>I is homeomorphic to S2 x S2. We show that the homology intersection properties of M and Mi are distinct, hence, proving that they are topologically different. A base for HiM, Z) is given by {SI' S2} where SI = to} X 1Jl>1, S2 = IJl>I X to}. Hence, any 2-cycIe C is homologous ("') to as, + bS 2 , a, b e Z. The intersection multiplicity I(C, C) = J(aS, + bS 2 , aS I + bS 2 ) = a 2 [(SI' SI) + b 2[(S2 ,S2) + 2abl(SI' S2). Since St. S2 occur as fibres in IJl>I x 1Jl>1, [(SI, S,) = [(S2' S2) = o. Hence, Proof

I(C, C) = 2ab

=0 (mod 2).

(1)

16

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

In M we have the following picture:

w'

w M

./"" ~

V

I--

.

s

s, Figure 6

where At" is the submanifold of M~ defined by C= c and C* = zc with the coordinates explained before. Then At" is a 2-cycle and Ao '" Ac. Hence /(Ao, Ao) = /(Ao, Ac) = 1. Since for any 2-cycle Z on M, /(Z, Z) == 0 (mod 2) we see M::f: Mr. REMARKS

M!::f: M:(m ::f: n) as complex manifolds. 2. M~m = M topologically. 3. M~m+1 = M~ topologically. These facts are proved in Hirzebruch (1951).

1.

EXAMPLE 2. (Logarithmic Transformation) LetM = T x Pl,T = C/G, G = {mw + n I m, n E 7L, 1m w > O} where T is a torus of complex dimension 1. For any CE C, we denote the class in C/G = T by [C]' We perform surgery on M as follows: Let S = {O} x T, W = D x T where p1 = C u {<X)} and

OED = {z E

Clizi < e}. T

w

s Figure 7

Then set W* = D x T = {z, [(*] I zED, [(*] T. Define/: W* - S*~ W - S as follows: /: (z, [(*]) ~ {z, [(*

where 0 <

Izi < e.

E

T} and S*

+ (l/2ni) log z]},

= {OJ x T £

D x

3.

CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS

17

Then f is biholomorphic and we can form M* = (M - S) u W*, where (z, [C]) = (z, [(*]) if [C] = [(* + (1/2ni) log z], 0 < Izl < 6. For the first Betti numbers b l we have b 2 (M) = bz(T) = 2, but b l (M*) = I. In fact, M* is topologically homeomorphic to S3 x sl.

REMARK.

~

7L ffi 7L is clear by the Kunneth theorem. To study axT where a is a closed disk, and T is homeomorphic to SI x Sl. If ( = x + yw, we can identify [C] with (x, y), where x + 1 is identified with x, y + 1 with y, where x and yare real (E IR). Therefore W* = D X Sl X Sl, M - W = a x Sl x Sl. Since we are only interested for the moment in the topological type Proof

H 2 (M, Z)

M*, first we notice that M - W = (l?1 - D) x T is homeomorphic to

of M* we may as well assume that D is the unit disk and that the identification in the definition of surgery takes place on the boundary of D = {e i9 10 $ () $ 2n}. Then we identify (w, x*, y*) and (w, x, y) if x = x* + «(}/2n), y = y*. Hence, M* = B X SI where B is a circle bundle over S2 ; and in fact, we easily see by the transition function that this is the Hopf bundle S3 -+ S2. Hence B = S3. This proves that M* = S3 X Sl; b1(M*) = 1 follows.

EXAMPLE 3. We mention also the classical quadric transformation (blowing up, u-process). First we discuss the case where M has complex dimension 2. Let S = p be any point on M, and let S* = pI be a copy of the Riemann sphere. We define M* = (M - p) u pi as follows: Choose a coordinate patch W = {(ZI' z2)llz11 < 6, IZ21 < 6} in a neighborhood of p so that Zl(P) = zz{p) = o. We define a submanifold W* of W x pI as follows:

W* = {(ZI' Z2; (I' (2) E W x pi I ZI(2 - Z2 (I = O}, where ((1(2) are homogeneous coordinates on pl. W* is a submanifold since (aflaz 1) = (2' (af/az z ) = -(I iff= ZtC2 - ZZ(l' and hence [(af/az 1), (af/az z)] :F (0, 0). Letf*: W* -+ W be the restriction of the projection map W x pi -+ W to W*. Then W* 20 X pi = S*, f*: S* -+p = (0,0), andf*: W* - S*-+ W - pis biholomorphic. The first two statements are obvious. For the proof of the last, let (ZI' Zz; (10 (z) f/ S*. Then at least one of Zi:f: 0 and hence «(I' (z) is determined by (Zl' ZZ)f*-I: (Zlo Z2) -+ (Zl' Zz; Zl, zz). By surgery we obtain M* = (M - p) u pl. We make the following definition: DEFINITION 3.4. The quadric transformation Qp with center p is the manifold Qp(M) = M*. REMARK.

QPm··· Qp,(pZ) can be complicated! For example, Qp6 ... Qp,(pz) = {( I(~

+ ci + ,~ + (~ =

O}

S;

p3.

18

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

For manifolds M of dimension

~

2 we proceed analogously. If

dime M = n, let (Zl •... , zn) be coordinates centered at pEp = (0, ···,0)]. If W = {(Zl' ... , zn)llz,,1 < I $ C( $ n}, we set W* = {(z, c) IZ;. C. - Z. C. . = O,i $ A, v $ n} s;;; W x pn-l. Again W* is a manifold, projection onto W defines a biholomorphic map W* - pn-l -+ W - p, by (z, 0 -+ z. We form M* = (M - p) u W* = (M - p) U pn-l and call M* = QP(M) the quadric transform of M with center p. E,

4.

Analytic Families; Deformations

Consider a torus Tro = CfG, G = {mw + n I m, n E 71. 1m w -+ O}. We have a family of tori depending on the parameter w. Many examples of compact complex manifolds depend on parameters built into their definitions. We also have the examples of hypersurfaces of degree d in pn. Each such surface Md = {C 1/<0 = O} is defined by a function I of the form 1= Lka+'.'+kn=d aka ••. k n 'io ... C~n. In a sense to be made precise Md depends" analytically" on the coefficients aka •.• k n off We make the following definition: DEFINITION

4.1.

Let B be a (connected) complex manifold and let

{M,l t E B} be a set of compact complex manifolds depending on t E B. We say that M, depends holomorphically (or complex analytically) on t and that {M,l t E B} forms a complex analyticlamily if there is a complex manifold .It and a holomorphic map (jj onto B such that (I) (jj-l(t) = M, for each t E B, and (2) the rank of the Jacobian of (jj is equal to the complex dimension of B at each point of .It. We note that (2) implies M, is a complex submanifold of .It. Now for some examples. As before, we denote Tw = CfG,

G = {n

+ mw I n, m E 71., 1m w > OJ.

Let B = {w 11m w > O} c: C. Let f§ = {9mn 19mn: (w, z) -+ (w, Z + mw + n)}. Then f§ is a properly discontinuous group of transformations on B x C without fixed point. Hence, .It = B x C/f§ is a complex manifold. The projection map B x C -+ B induces a hoi om orphic map .It ..!! B, and (jj-I(W) = Tw. It is easy to see that the Jacobian condition is satisfied so {T", I wEB} forms a complex analytic family. But suppose we proceed as follows: Again Tw = CfG and the map C -+ CfG is written Z -+ [z]. Let D = unit disk = {tlltl < I}. On D x Tro consider the group f§ = {I, 9} where 9: (t, [z]) -+ ( - t, [z + !]) is of order 2.

4.

ANALYTIC FAMILIES; DEFORMATIONS

19

Then I'§ is properly discontinuous and has no fixed points so D x T(J)/I'§ is a complex manifold. Let 1t: D x T(J) -+ D be defined by (t, [z]) -+'t = t 2 • Then the diagram

(t, [z]) ~ (-t, [z

.j " t2

__

+ t])

j.

t2

commutes so 1t defines a holomorphic map on .;It. The Jacobian condition is not satisfied by 1t, since (j'J't/ot) = 2t = 0 at t = o. We notice that 1t- 1('t) = T if't =1= 0, but 1t -1(0) = T*, a torus of period w/2. (J)

DEFINITION 4.2. Let M, N be compact complex manifolds. M is a deformation of N if there is a complex analytic family such that M, N s;;; {M t It E B}, that is, M ta = M, Mtl = N. We have the following sequence of problems to guide our work: PROBLEM.

Determine all complex structures on a given X.

PROBLEM.

Determine all deformations of a given compact manifold

PROBLEM. of a given M.

(easier?) Determine all "sufficiently small" deformations

M.

DEFINITION 4.3. We say that all sufficiently small deformations have a certain property f!jJ if, for any complex analytic family {M t I t E B} such that M ta = M, we can find a neighborhood N, to ENe B such that M t has f!jJ for each tEN. By standard techniques in differential topology we prove the following theorem: THEOREM 4.1. Let M t be a complex analytic family of complex manifolds M t • Then M t and M to are diffeomorphic for any t, to E B. Proof The reader will notice that we really only use the differentiability of the map 1t: .;It -+ B, analyticity is not needed. In fact, we prove: Let .;It be a differentiable family of compact differentiable manifolds such that the differentiable map 1t: .;It -+ B has maximal rank (.;It and B are differentiable manifolds). Then M t is diffeomorphic to M ta •

20

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

First we construct a Coo vector field 0 on a neighborhood of M,o in ..it such that 1t induces 7t*(0) = a/as, where s is a member of a coordinate system (s, x 2 , ••• ,xm) in a neighborhood of the point 10 e B chosen as follows:

Figure 8

We connect 10 and t by an embedded arcy: (-e,1 + e) -+ {yes) Is e( - e,1 + e)}. A compactness argument shows that we can assume that I and 10 lie in the same coordinate patch and since y is an embedding we can find a chart with coordinate (s, 12 , ••. ,1m) around lo(to = (0, ... , 0), I = (s, 0, ... , 0». Because of the rank condition, 1t- 1 (y) = 7t- I {(s, 0, ···,0) I -e < s < 1 + e}, is a submanifold of ..it, and we can assume that (s, xf, ... , xj) are coordinates of ..it for a given point of 7t - I (y) in some neighborhood qJJ of the point. Then the vector field (a/as)j on qJj satisfies 1t.(a/as)j = a/as. Then if {Pj} is a partition of unity subordinate to {OIl j} (uOU j is a neighborhood of M,o)' the vector field 0 = LJ pj(a/as)j satisfies our requirements. For the second part of the proof we seek a solution of the differential equation d

(1)

ds xj(r) = 0j[x(r)],

where 0j is the a-component of 0 in the coordinate patch qJ j ' with initial conditions xj(O) = y", where (0, y2, ... , y") is some point close to (0, ···,0). If s is small enough and Iyl is small enough, Equation (1) has a unique solution xj(r, y) on some small interval. By compactness, we can assume that M,o c: U jqJj' a finite union of such patches, and that in each qJj' (1) is satisfied for Irl < jJ. where jJ. is independent ofj. If xj(r, y) is such a solution, let Xj = jjk(Xk) and define ef[r,hiO, y)] uniquely on qJj (\ qJk by (2) Then dxj(r, y) =

dr

L axj aef[r,jkj(O, y)] /I

axf

or

'

4.

ANALYTIC FAMILIES; DEFORMATIONS

21

and by the uniqueness of the solution to (I)

x/I:, y) = jjk(Xk[T,f,.iO,

y)]).

(3)

Equation (3) implies that xC-t", y), ITI < p, y E M ,O is a well-defined differentiable map defined on M,O for each T, It I < p., and x(O, y) = y. Let cpiy) = x(t, y); then CPo = id (on M,o )' It is also easy to check that 1t[cpt(Y)] = yet) since 1t.(0) = dlds. Hence, CPt maps M,O into My(t) (for small t). We can repeat this argument for My(t) and define t/lv: My(t) -+ My(t+v) and by uniqueness get t/I - t 0 CPt = id, CPt 0 t/I _to = id. Since everything is differentiable, the theorem is proved. Q.E.D. REMARK. This argument is very old. For a treatment from the point of view of Morse theory, see Milnor (1963). Sometimes this theorem is attributed to Ehresmann (1947). We consider some more examples of complex analytic families. The dependence of the complex structure of M I on t E B can be complicated as we shall see.

EXAMPLE I. Consider again the family of tori {Teo I w E H} where H = {w 11m w > O} and Teo = CjG, G = {mw + n I m, n E Z}. From the classical theory of Riemann surfaces we see that Teo and Teo' are conformally equivalent if Wi = (aw + blew + d) where a, b, e, dE Z, and ad - be = l. Let r§ be the group of transformations acting on H which have the form w-+

aw + b , ew + d

a, b, e, dE Z,

ad - be = l.

Then it is easily seen that r§ is properly discontinuous on H. A fundamental region IF for r§(ug§ = H, g§ n IF = cP if g ::f id) is given by the shaded region in the figure below, hence Teo ::f Teo" if w ::f Wi and w, Wi E IF. 1

Figure 9

22

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

The elliptic modular function J defines a conformal map J: H/~ -+ C. So Tw = Tw' if J(w) = J(w'). EXAMPLE 2. (n-dimensional tori) We give an outline of some of the facts. A torus Tn = Cn/G where G = {L7~ 1 miwi I mi eZ} where WI'···' W211 are complex n-vectors linearly independent over R

(a) We can replace {wi} by any other linearly independent basis of G. That is, 2n

wi).

= k=l L aikwU,

(4)

where aik e Z, det (a ik ) = 1 are also permissible generators of the lattice (group) G. (b) We may also introduce new coordinates in cn so Z). -+ 2)., where n

2l

=

L ZvYvl, Yvl e C, v=1

Then, (5) The resulting change from Equations (4) and (5) becomes (6) We may assume that W n + 1 , ••• , W2n are C-linearly independent. Hence by some change of coordinates (Yv).), we can obtain

... w) (6)11 In (Yv).) = 6>nl

.•.

Q)ln) Wnn ,

W

2nn

(7)

I

where I is the n x n identity matrix. So we may assume (wij)

= (~),

where

n = (wij) 1 ~ i, j ~ n

(b···~)· (c)

We can also break (a ik ) into pieces:

Then (4) takes the form

wi). =

(ajk)(~) = (gD,

n~ =

An + B, n~ = cn + D.

and 1=

4. If one assumes that

23

ANALYTIC FAMILIES; DEFORMATIONS

0; is invertible, then (~D(O;)-I = (~') where 0' = (AO

+ B)(CO + D)-I,

det(~ ~) =

(8)

1.

The following treatment will be a bit sketchy; for more details consult Kodaira-Spencer II (1958). The fact that WI' " ' , W n , (l, 0, "', 0), (0, I, 0, 0 ,), ... (0, ... , 0, I) are real linearly independent implies

det(~ ~) t= 0, which is the same as (2it det [Im(wJ).)] t= o. Consider the space H = {O det(Im 0) > O} [some sort of a generalization of 1m W > 0 in Example (1)]. Let C§ = the set of all transformations 0-+ (An

where

+ B)(CO + D)-I = n',

(~ ~) E SL(n, I), the invertible integral matrices of determinant + I.

This group does not really act on H since it is possible for CO + D to be singular; one should consult Kodaira-Spencer for more details. H should be extended to something more general on which SL(n, I) acts. In any case,

Tn= Tn-,

if 0' = gO, 9

E C§.

We would like to form H/C§. But it turns out that C§ is not discontinuous. In fact, for any open set U c H, there is a point n E U such that {gO Ig E 'Y} n U is infinite. Hence, the topologial space H/C§ with the quotient topology is not Hausdorff and hence certainly not even a topological manifold by the usual definition. We next give some examples of families {M t 1 t E B} such that M t = M for t t= to and M to t= M. EXAMPLE 3. A Hop! surface is a compact complex manifold of complex dimension two which has W = (:2 - {(O, O)} as universal covering surface. More precisely, the Hopf surface M t is defined by M t = WI Gt where Gt =

{gm I mEl} and g: (ZI' Z2) -+ (azl where 0 <

LEMMA

lal < I and

t

{M t It

q

4.1.

E

E

+ tz2 , aZ2)'

that is, (::)

-+

(~ :)(:~),

C. Then M t is a compact complex manifold.

is a complex analytic family.

24

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS Proof

= {M t I t E C} = C x W/f, where f = {ym I mE Z}, and

y(D (~ =

i DeJ

Q.E.D.

We claim (1) (2)

M, = Ml (complex analytically) for t #= O. Mo #= MI'

Proof of (1).

We make the following change of coordinates:

Then the equation

implies that M 1 Proof of (2).

= M t when t #= O. First we prove a special case of Hartog's lemma.

LEMMA 4.2. Any holomorphic function defined on W = C 2 be extended to a unique holomorphic function on C 2 • Proof Let f(zl' define the function

Z2)

-

{CO, On can

be the function on W. Pick a number r > 0, and

1 1. few, 2m Iwl=r w -

F(Zl' Z2) = - . j

Z2)

dw,

ZI

for Izd < rand Z2 arbitrary. Then F(zl' Z2) is an analytic function in its cylinder of definition which is a neighborhood of (0, 0). If we can prove f = F where both are defined, we will be finished. We know thatf(w, Z2) is holomorphic if Z2 #= O. So Cauchy's theorem gives

Fix Zl' 0 < Izd < r. Then F(ZI' in Z2; therefore,

Z2)

= f(zl' Z2) for

Z2

#= O. Both are analytic

F(zl' 0) = f(zi' 0).

Hence they agree where defined, proving the lemma. Now let us suppose M, = Mo. I oF O. Then there is a biholomorphic map f: M, -+ Mo. W is the universal covering manifold of M, and Mo. sofinduces

4.

ANALYTIC FAMILIES; DEFORMATIONS

25

a map I: W --t W which is biholomorphic, such that

W~W

G'l

I

Go

f

Mt---+M o commutes. It follows that Gt = 1-' Go! Hence for generator 9t of Gt ,

9, =1-'9"5' f.

(9)

Write the map I in coordinates as

I(z" zz) = [J,(z" zz)'/z(z" Z2)]. Then by Hartog's lemma extend liz" zz) to a holomorphic function F;. (z" zz) on C Z• Then F maps C Z into C Z [F = (F1 , F2 )], and F(O) = O. For if not, extend 1-1 to F which satisfies F[F(z)] = z on Wand by continuity, F[F(O)] = O. But if F(O) =F 0, £[F(O)] = I-I [F(O)] =F O. This contradiction gives the result. Now expand F)"

F;,(ZI' zz) = F)"z, We know thatf[9,(z)]

+ F).,zz + F;'3ZT + F).• ZIZ2 + ....

= 9"5 I [f(z)] so F[g,(z)J =

(~ O)±I rx F(z).

Rewriting this gives

F,(rxz, + lZ 2 , rxZ2) = rx±IF,(ZI' zz), Fz«(1.z1 + lz 2 , cxzz) = rx±IP1(ZI, zz)· Expanding these and taking the linear terms yields

(P P Il

li

t)

0) (PF

P'z) (rx = (rx P1.2 0 rx 0 rx

±1

ll l ,

This can only happen when t = O. Hence M 1 =F Mo.

Q.E.D.

EXAMPLE 4. Ruled Surfaces (examples of surgery) Our ruled surfaces will be IFDI bundles over IFD'. Let IFDI = {' I' E C U {oo}} (nonhomogeneous coordinates). M(m) = VI x 1FD1 U V l x.1FD 1 where VI u V l = IFDI, VI = C, V l = 1FD1 - {O}, and identification takes place as follows (recall Section 3): Let (ZI' (I) E VI x 1FD1, (Zl' ~2) E V l X IFDI. Then

REMARK.

MC",)

=F

M(I)

for m =F t' (not to be proved now).

26

DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS

THEOREM 4.2. M(t) is a deformation of M(m) if m - t = 0 (mod 2). Assume that In> f. Then there is a complex analytic family {M t I I E C} such that Mo = M(m) and M, = M(f) for 1:1= o. Define M, as follows: M, = VI X pI U V 2 X pI where (ZI' (I) if ZI = l/z 2 , (I = Z~(2 + tz~ where k = !(m - t). Then it is easy to see that {M, t E q is a complex analytic family and that Mo = M(m). Suppose 1:1= O. Introduce new coordinates on the first PI by Proof

+-4(Z2' (2)

(' _z~' 1-

I -

1(1

t (linear fractional transformation).

On the second pi, r'

'2 +

'>2 = I '" kv 22

Then, using

ZIZ2 =

I, and

(I =

Hence, in the new coordinates,

Z~'(2

+ IzL

ZI Z2 =

for PROBLEM. such that

C,2

t2·

we get

I, (~ = z£(~; so

t:f= O.

Q.E.D.

Finda pair ofcomplex analytic families

(a)

Mo

(b) (c)

M, = No N, = Mo

=1=

No, for t for t

=1= =1=

0, O.

{M,lltl <

(not complex analytically homeomorphic)

There are no known examples of this type.

I}, {N,II/I < I}

[2] Sheaves and Cohomology I.

Germs of Functions

Let M be a complex (or differentiable) manifold. A local holomorphic (differentiable)function isaholomorphic (differentiable) function defined on an open subset U £; M. We write D
rx,

p E C,

fp' gp = (fg)p, are well defined, hence,

(!)p, £1)p

become linear spaces over IC. We also define,

We put a topology on (!) and El) as follows: Take any cp E (!) (or El); then cp E (!)p (or El)p) for some p. Take any holomorphic (differentiable) 1 with Ip = cp and define a neighborhood of cp as follows:

where p E U £; M, U is an open set in D(f). It is easy to see that the system of neighborhoods (il1(cp;f, U) defines a topology on (!) (or £1). EXAMPLE. (!) on the complex plane C. Let p E IC. Then if 1 and g are holmorphic at p we have expansions valid in some neighborhood of p, co

fez)

= L fk(z

00

- pt, g(z)

k=O

= L gk(Z -

pt,

k=O

so 1 and g are equivalent at p if and only ifIk = gk for all k. Hence, the germ at p is represented by a convergent power series; (!)p = ring of convergent power series. And an element cp E (!)p can be represented by cp = Ip = {p;/o ,fl' ... } where Iimk .... oo I/kll/k < + 00 and the radius of convergence is r(cp) = 1/ lim. 27

28

SHEA YES AND COHOMOLOGY

We define 0fI{(J);

E)

= {t/I I t/I = I q , Iq - pi < E where 0 < E < r{(J)}.

In terms of our representation we calculate 00

I(z)

00

= L Ik(Z

- p)k

k;O

=I

fm{Z _ q

+ q _ p)m

111;0

Hence 0fI«(J);

E) =

{t/I I t/I = (q; go,"', gk" .. ), I(q -

gk =

m~k (;)fm(q -

p)1 <

E

p)m-k}.

We note that t/I E d//{(J); E) means that t/I is a direct analytic continuation of (J). The case of ~ on IR is not so simple. If (J) = Ip where I is a Coo function atp, III

j(x) =

I

fk(X - p)k

+ O(x _

p)m.

k=O

But I is not determined by the Ik'S since there exist COO functions I which are not identically zero, but which have all derivatives zero at some point. Define w: (!) (or ~) -. M by w«(!)p) = p. PROPOSITION 1.1. (1) wis a local homeomorphism (that is, there exists 0fI such that w: 0fI«(J);f, U) -+ U is a homeomorphism). (2) w-l(p) = (!)p (or ~p) (obvious). (3) The module operations on w-l(p) are continuous (that is, IX(J) + IN depends continuously on (J), t/I).

Proof (1) 0fI«(J);f, U) = {fq I q E U} and w: /q -+ q is certainly 1 - 1. It is obvious that w is continuous. To show that w- l is continuous, let OfI(w; g, V) be a neighborhood of t/I = f 4 . We want to find a neighborhood W of q so that fw=W-I(W)EOfI(t/I; g,V) for wE W. We know that 9q =t/I =Iq , so / and g are equivalent at q. Hence, 1= 9 in some neighborhood N of q. Let W = N n V. Then/w =gw on W, so Iw E 0fI(t/I; g, V) for wE W. This proves that the w- l is continuous. (3) Let (J)=/p,t/I=gp' Then 1X(J)+pt/l={~f+pg)p. Let OfI{IX(J)+pt/l; h, U) be a neighborhood of IX(J) + Pt/l. Then IX(J) + pt/l = hp = (IX! + pg)p so

1.

h =rxf + flg in some neighborhood V£ U of p. Then if • E all(1/1; g, V), we have rxu

+ fl.

+ fll/l;

U E

OJI(lp;J, V),

rxfq + flgq = (rxf + flg)q

=

= hq E OJI(rxlp Since OJI(rxlp

29

GERMS OF FUNCTIONS

h, V) £ O//(rxlp

+ fll/l; h,

+ fll/l;

h, V).

U) we are done.

Q.E.D.

We now give a formal definition. Let X be a paracompact Hausdorff space. 1.1. A sheaf Y over X is a topological space with a map Y --. X onto X such that

DEFINITION

w:

(1) iii is a local homeomorphism [that is, each point s E 9' has a neigh. borhood all such that w: OJI -+ w(OJI) c X is a homeomorphism onto an open neighborhood of w(s)]. (2) iii-1(x), x E X is an R-module where R = 71., IR, C, or principal ideal ring. (3) The module operation (s, t) -+ rxs + flt is continuous on w-1(x) where rx, fl E R.

(The reader can easily generalize this definition, but for our purposes it suffices.) The set Y" = w-1(x) is called the stalk of Y over x.

EXAMPLES. (of sheaves) on a complex manifold. (2) ~ on a differentiable manifold. (3) The sheaf over X of germs of continuous (Ill or C valued) functions. (4) The sheaf over X of germs of constant functions. (l)

(!J

In Example (4) 9' = X x C with the following topology: Let s = (x, z); then OJI(s) = {(y, z) lyE U, z fixed}. If r -+ f(r) is a continuous map into Y of 1= {r Ia < r < b}, then f(l) = {(y, z) I z fixed and y = w(f(r»r E l}. In other words we give X x C the product topology where X has its given topology and C has the discrete topology. DEFINITION 1.2. Let U be a subset (usually open) of X. By a section u of 9' over U we mean a continuous map x --. u(x) such that iii u(x) = x. Suppose X = M, a complex (or differentiable) manifold; and suppose Y = (!J (or ~). If f(z) is a holomorphic (or differentiable) function on U, then u: p -+ f p , p E U is a section.

30

SHEAVES AND COHOMOLOGY

1.2. Let (1: V ~ f/ be a section (f/ as above). Then (1 determines a holomorphic (or differentiable function) 1 = I(z) on V such that

PROPOSITION

a(p)

= Ip.

Proof a(p) E (!)p (or ~p). Hence there is a holomorphic (or differentiable) g(z) defined on some neighborhood of p so that a(p) = gp. Since g depends on p we write, g(z) = gCpl(z). Define 1 as follows: I(p)

= g(P)(p).

Then 1 is obviously well defined. Then (1) I(p) is a holomorphic (differentiable) function on V.

Take Wa neighborhood of p, W ~ V. Let dIJ = dlt[a(p); g(Pl, W] = {(g(P\ Iq E W}. Since a is continuous, for any small neighborhood N of p, N ~ W, we have a(N) ~ dIJ. Hence a(q) = (g(P\ . But we also know a(q) = (g(q»q. Thus, (g(q\ = (g(P\, and g(q)(z) = g(Pl(z) for z in a small neighborhood V of q, V c N. But I(q) = g(q)(q) = g(p)(q) for q E V. So I(z) = g(P)(z) for z E V and g(P) holomorphic (or differentiable) in V implies that 1 is also. Proof

(2)

By definition a(p) = (g(Pl)p = Ip for each p

E V.

Q.E.D.

Hence we have the maps: local holomorphic (differentiable) functions

t germs

t sections = holomorphic (or differentiable) functions.

re

V, f/) will denote the R-module consisting of all sections of f/ over V. We remark that V, (!) are all holomorphic functions over V and V,~) are all differentiable functions over V. Let {V ;.11 ~ A. ~ n} be a finite family of open sets in X such that n V). ::f: ljJ. Let a). E V)., f/) and IX;. E R. Then L IX). a). E reV, f/) where V = n V).. Let W be an open set and a E V, f/) for some open set V. Then x -+ a(x), x E W n V defines a section of W n V, f/). We denote this section by rwa and call it the restriction of a to W n V.

re

re

rc

rc

2.

rc

Cohomology Groups

Let X be a Hausdorff paracompact space and let f/ be a sheaf over X. Fix a locally finite covering Ii/i = {V j } of X. A O-cochain CO on X is a set CO ={aj} of sections a J E reVj')' A 1-cochain C· = {ajd is a set of sections

2.

COHOMOLOGY GROUPS

31

Ujk E qUj n Uk, f/) such that Ujk = -Ukj (skew-symmetric). A q-cochain q = {u jo '" A} is a set of sections u jo "'ik E qujo n ... n U j .' f/) which are skew-symmetric in the indices jo ... A. Let Cq(Olt) be the R-module of all q-cochains. We define a map Cq(0lt)...!..Cq+1(0lt), the coboundary map as follows: For O-cochains, t5Co = {Tjk} = {Uk -uJ where CO = {ud; for l-cochains C 1 = {ujd, t5C 1 = {Tjk/} where Tjkl = Uu -u jl + u jk = Ujk + Ukl + Utj . In general, t5C q = {Tjo ... j.+.} if Cq = {Ujo ... j.}, where

c

+ (_l)q+lu jo "' j ' = L (-l)kujo ··· j~ ... i.+.' where

1\

(1)

means "omit."

We denote the q-cocycles by zq(Olt) = {C q I t5C q =O}.

The q-cohomology group (with respect to Olt) is

(2) We should remark that t5C is always skew-symmetric and t5t5 = 0 so that t5Cq-l(Olt) £; zq(Olt) and Equation (2) makes sense. The qth cohomology group of X with coefficients in the sheaf f/ is defined to be

Hq(X, f/)

= lim Hq(Olt, f/). 'II

This limiting process will now be explained. We say that the open covering "Y = {V;J AeA of X is arefinementofOlt = {U JieJ if there isamap s: A -+ J such that VA c U.(A) = Uj(A) , where we setj(A) = SeA). We define a homomorphism n~

: (q(OU) -+ U("Y),

nt: {uio·"j.} -+ {TAO'" A.}' where

(3) It is easy to check that t5n~ = n~ t5,

(4)

so that n~ maps zq(°lt) into zq("Y) and t5C q- 1(fI) into t5Cq-l('f"). Hence n~ induces a homomorphian n~: Hq(:5II) --+ Hq('f"). LEMMA 2.1. n~: Hq(:Jlt) -+ W('f") is independent of the choice of map s : A ...... J in the definition of refinement.

32

SHEAVES AND COHOMOLOGY

Proof First some notation: fix indices V = V;Clo (')... (') V;CEq , VI

= V;

(10

./"-... Uil = Uf(ao) (') ... (') Uf(aj) (') Ug(a j )

A. Let

IXo, ••• , IXq E A.

(')... (') V;fl.1 (')... (') V;IZq , (') • • •

./"-... U9(II/) (') ... (') Ug(aq) ,

and Ui

=

Uf(II,) (') ... (') Uf(IIj) (') Ug(II J )

(') ••• (')

Ug(II.) '

where!, g: A ~ J are two refining maps. Define a function (kU)A, ... A. by q

(kuh, ... A. =

p"f;o ( -1)P- 1 rV ouf(A,)· .. f(Ap)g(Ap)·· ·g(A.)

(5)

Let us call the maps n~, defined by f and g,f*, and g*. We claim that the following equation holds: [(ok

+

(6)

kO)]IIo ... a. = (g*u - 1*u)ao ... a•.

The function ku is not necessarily skew-symmetric in its indices; so we skew-symmetrize

-r~I .. · A = (k'-r)A' ... A =~, L sgn(Al •

•

q.

P.I

Next we use (6) to see that [(15k'

+ k'b)u]ao ... aq =

(g*u - 1*u)IIo ... II•.

Hence, if bu = 0, bk'u = g*u - f*u E bCq- 1(1'). Hence,f* and g* induce the same map, Hq(CJlt) ~ Hq(1'). Therefore we prove (6). The reader can easily check the following calculations: q

(bku)ao ... II = •

L (-1)'ry(ku)w··;, ... a. 1=0

=

t (- [tt\ + t (1try

{=o

i=t+

(bku)aO···II. =

i=O

l)i ryt u f(IIo)··· f(IIi9(IIi) ... g{;;) ... g(a.)

l)i - 1 r v{ u f(IIo) ... ;(a';)f(IIJ)g(IIJ) ... g(a.)]

1

~ (-1)(+iryu/(IIo)···f(IIj)g(aJ) ... ~) ... g(a.) )
' + j>t L...( -

1){+i+ 1 rv u /(.0) ... f(II,) ... f(IIj)g(II ) ... g(II ). J

(7)

q

Similarly, (kbu)IIo···II = I,(-1)j+tryuJ(IIo)···;(a';)···f(II·)9(IIJ) ··g(II) q tSj J q

(8)

COHOMOLOGY GROUPS

2.

33

Equations (7) and (8) give

q

- L ry

0j(IIo) "'/(IIj)g(IIj) ... g(IIq)

j=O

= ry O"g(IIO) ... g(IIq)

-

rV 0"/(110) "'/(II q) '

(9)

Q.E.D.

proving Equation (6).

Knowing that the map n~ depends only on 1111 and "Y, we proceed to the definition of the limit. We write 1111 < -H' if -H' is a locally-finite refinement of 1111. Then < is a partial order and given 1111, "Y there is -H' so that 1111 < -H' and "Y < -H'. Hence the set of all locally finite coverings of X forms a directed set with respect to <, and the following equations can be verified (using Lemma 2.1):

n: = id, n:,. =

DEFINITION 2.1.

n~

0

n~,

Hq(X,!/) = lim Hq(l1I1, !/).

'" REMARK.

We recall the definition of the limit lim. We say that g, hE Hq

'" n:,. 9 = n:, h. Denote

(1111, !/) are equivalent if there exists -H' > 1111 such that the equivalence class of 9 by g. Let

Hq(I1I1,!/) = {g \g E HQ(I1I1, !/)}. The map 9 ~

9 defines

a homomorphism II"', n~

: HQ(I1I1, !/) ~ HQ("Y, !/),

and n~ induces a homomorphism n~,

_.'" : HQ(I1I1, 11.,. !/) ~ HQ("Y, !/). n~ is injective.

LEMMA 2.2.

Proof

n~g = 0 if and only if

o and 9 = O.

n:,.

0

n~ 9 = 0 for some W. So

n:,. 9 =

Q.E.D.

Hence, identifying H4(11I1,!/) with n~H4(11I1, !/), we may consider HQ(I1I1, !/) c H4("Y, !/) provided that 1111 < "Y. Then by definition, H9(X, !/) =

UHQ(I1I1, !/),

'"

34 and

SHEAVES AND COHOMOLOGY

n'1': Hq(lJIt, f/) - t Hq(lJIt, f/)

£;;

Hq(X, f/) is a homomorphism of

Hq(lJIt, f/) into Hq(X, f/). PROPOSITION 2.1.

Proof

HO(X, f/) =

rex, f/).

= 0 so HO(IJIt, f/) = ZO(IJIt, f/). f/) = [0'10' = {O'j},O'j E r(Vj' f/), DO' = OJ.

By definition C- l

ZO(IJIt,

But (j0' = 0 means O'j(z) - O'k(Z) = 0 on Vj n Vk . Hence O'(z) E r(X, f/), defined by O'(z) = O'j(z) when Z E V j , is meaningful. This proves HO(IJIt, f/) = r(X, f/) and implies HO(X, f/) = r(X, f/). Q.E.D. PROPOSITION 2.2. COROLLARY.

H"": HI(IJIt, f/)

HI(X, f/) =

-t

HI(X, f/) is injective.

U HI(IJIt, f/). ""

Proof (of the proposition). Suppose hE HI(IJIt, f/) = ZI(IJIt)/DCO(d/t). Then h = {O')k}' O'jk E r( Vj n Vk , f/) where 0' ij + O'jk + O'ki = O. We want to show that n""h = 0 implies h = O. n""h = 0 means Ii = 0 and this is true if and only if n~h = 0 for some 1', l ' > 1JIt. Let 1(/ = {WjA I WjA = Vi n V).}. Then "If/" is a locally finite refinement of l ' and n~h = n~ 0 n~h = O. Also 1(/ > d/t since 1(/ i). C Vi and we can use the maps(iA,) = ; in the definition of refinement. Then we have where 't(j).)(jll)

=

'tjAjll

= fW,.l.'" Wj,. O'ij'

Then n~v h = 0 implies {'t i).jll} = D{ 't i).}' that is, 't i).jll = 't jll - 't i).' Since 'tWIl = rW •.l.",W.,.O'ii = 0, we obtain 'till = 'ti). on Wi). n Will' Vi = U).WjA, and 't i = 't il' on Will defines an element 't i E r( Vi' f/). Then the equation 0' ij = 't j 't i implies h = O. Q.E.D. Consequently, in order to describe an element of HI(X, f/), it is sufficient to give an element of HI(IJIt, f/) for some 1JIt.

EXAMPLE. dime Hl(M, l!J)

Proof.

Let M = {(Zl' z2)llzll < 1, IZzl < 1, (Zl' Z2) =F (0, O)}. Then

= + 00.

Set

VI = {(ZI' zz) I (ZI' zz) E M, ZI =I:- OJ,

VZ = {(Zl' zz) I (Zl' Z2) E M, Z2

=I:-

OJ.

3.

INFINITESIMAL DEFORMAnONS

35

In this case M = UI U U z so chose as covering 0/1 = {VI' U z }. Then HI(o/1, (9) = ZI(OlI, (9)/bCO(o/1, (9) where ZI(o/1, (9) = {0'121 0'12 E r(VI () U z , (9)},Co(o/1, (9) = {t It = (tl' tz), tit E r( V"' (9)}, and bCo(o/1, (9) = {tz - tl Itit E r(V", (O)}. We note that VI () U z = {(ZI, zz) 10< IZII < 1,0 < IZzl < I}, so we have a Laurent expansion for 0'12

m=-CX)n=-co

tl IS holomorphic on VI = {(ZI' zz) 10 < IZII < I, IZzl < I} so tl(Z) = L~~ -ooL:'=obm"z/~z~. Similarly for tz, tz(z) = L~=oL:=OO_oocm"z~z~, and tz - tl = Lm~oor"2:0 am"z~z~. Then HI (0/1) ~ {0'121 0'12 = L;;;! -00 L;:! - 00 am"z'~z~}. Hence dim HI (0/1, f/) = +00 and since HI(o/1, f/) £;; HI(X, f/), dim HI(X, f/) = + 00. Q.E.D. PROPOSITION

2.3.

HI(X, f/) where d/I

If HI( V j, f/) = 0 for all Vj E 0/1, then HI(o/1, f/) ~ = {VJ.

Proof. We already know that HI(o/1, f/) £;; Hl(X, f/). Hence we only need to show the following. Let "Y = {VA} be any locally finite covering. Let if" = {WjAI WjA = Vj () VA}' Then it suffices to show that n:" : HI (0/1) -+ Hl("/Y) is surjective. Take a I-cocycle {O'jAb} of HI("/Y) where O'jAjlt + O'j"kv + O'k,jA = O. Then {O'WIt} for each fixed i is a I-cocycle on the covering {Wj)J of Uj' Since HI(Vj, f/) = O,HI({WU},f/) £;; HI(V j , f/)givesHl({Wu}, f/) = 0 for each i. This implies the existence oft iA E r( Wu , f/) such that aWIt = tjlt tiA' Let t be the O-cochain {tiA} on "/Y. Then {aIAh} = {aUk,} - bt defines a I-cocycle on "/Y which defines the same cohomology class in Hl("/Y) as 0'. From the definition of t we see that 0'1J.i1t = O. So O'iAj" + O'iltkv + O'~,jA = 0 yields O'lltkv = alAh' Similarly, O'jAh = O';ltkw' Hence, O'ik = aiAkv = ai,kv' and O'ik E r( V j () Uk, f/). Now we have found aik so that n:.(O'tk) = O'tU" and {aIAkv} is cohomologous to {ajAkv}' Hence is surjective. Q.E.D.

n::,.

3.

Infinitesimal Deformations

Using cohomology groups we will give an answer to the following problem: Let .;II = {M 1ft E B} be a complex analytic family of compact complex manifolds M I and let t = (tl, ... ,t") be a local coordinate on B. The problem is to define (aMI/at'). For this we define the sheaf of germs of holomorphic vector fields. Let M be a complex manifold and let W be an open subset of M. Let 0/1 = {V j, Z j}

36

SHEAVES AND COHOMOLOGY

be a covering of M with coordinates patches with coordinates p --+ Zj(p) = [zl(p), .. " z7(p)]. A holomorphic vector field () on W is given by a family of holomorphic functions {OJ} on W ('\ V j where n

0=

a

L OJ(p)-IX aZ

IX=I

j

on W ('\ Uj • These functions should behave as follows: On W ('\ U",

a L Of(p)p' 1 az" n

0=

(1=

We want

so the transition equation (1)

should be satisfied on W ('\ Uj ('\ U". Thus we have a definition of local holomorphic vector fields and we can define germs of local holomorphic vector fields. As notation we denote by 0 the sheaf over M of germs of holomorphic vector fields. (Later we shall give a formal definition of the holomorphic tangent bundle of a complex manifold.) Next we want to define the infinitesimal deformation (aM,/at.). First we consider the case B = {tlltl < r} £; C. .I{ is a complex manifold and iij: .I{ --+ B is a holomorphic map satisfying the usual conditions

M, = i.ij-I(t); (2) the rank of the Jacobian of iij = 1 = dim B. We can find an e > 0 small enough so that iij-I(A), A = {tlltl < e} looks as follows: (1)

J

iij-I(A)

= U OU j j= 1

(a union of a finite number of open sets).

On each OU j there should be a coordinate system

p --+ [z}(p), .. " zj(p), t(p)], where t(p) = iij(p) and such that OU j = {pi Izj(p)1 < ej. It(p)1 < e}. We write p = (Zj' t) = (z}, ... , zj, t). This construction is possible because rank iij = 1 These charts are holomorphically related so

zj(p) = fj,,[z~(p), . ", z~(p), t(p)] = fj"(z,, , t) on Uj

('\

Uk' Let U'j = M, ('\ OU j , It I < e. Then set

{(z} "', zj, t)llzjl < ej } =

V'j'

3.

37

INFINITESIMAL DEFORMATIONS

so we can use {(Z], ... , zj)llzjl < eJ as coordinates on VIj. The transformation zj = IMzk' t) depends on t. Consider p E dII j n dII j n dII". Then p = (Zj' t) = (Zj' t) = (Zk, t). So z~ = IMzk' t) = Jfj(Zj, t) = lil/ji z), t], where !jk = (fA .... .Jj,,). We set (). ( Jk

P. t

)= ~ ~

01=1

ajj,,(Zk, t) ~ :It :l 01· u UZj

Obviously (}j,,(t) E r( V'j n V,,,, 0,) where 0, is the sheaf of germs of holomorphic vector fields on M, .

Proof Before beginning the proof we remark that () ij + () jk + (}kj = 0 and (}ij = -(}jj is equivalent to (}j" = (}jj + OJ,,. To prove the lemma we differentiate the transition equation to get

t

an" = ar~ + arj af~k at at P=1 az~ at Then

(}." = L an" ~ = L af~j ~ + L af~k ~ I

01

at azQI

QI

at azi

(J

at

az~

Q.E.D. *DEFINITION 3.1. (dM,ldt) = {OJ,..(P' t)} E HI(M" 0,). We have made several choices in this definition and we must justify them.

PROPOSITION

3.1.

(dM ,Idt) is independent of the choice of local coordinate

covering {zj}.

Proof Let {r.J be a locally finite refinement of {dll j } such that are coordinates on r;. where

(C~.

t)

r;. = {(C;., t)II'~1 < e;.. It I < e}. Since {r;.} is a refinement of {dII j} we have a map s: A . . .+-J such that "f"). £; dII s ().). We also have holomorphic transition functions C{J).v where (~=

C{J;..«(., t)

on

r). n

Then the cocycle defined by this covering is

at a

"aC{J~. fT)..(t) = ~ a,~·

r •.

38

SHEAVES AND COHOMOLOGY

As before s induces a map s* : {Ojd

0A.(t) =

r1'";.n1'"v n

~

{OA.}, where

M.[Ojk(t)],j = S(A), k = s(v).

We must show that {Ih.} is cohomologous to {O;..}; that is, there exists a cochain {Oit)} such that Ihllt) - 0dt) = 0.(1) - Oit).

Since "Y;. c;;;; o/ij, j = S(A), there is a holomorphic gj such that zj "Y;.. The following equalities are clear: gj[IP;..(C., t), t]

= gj«(A' t) on

= gj(C;., t) = zj =

fMzk' t)

= fjk[gk«(;" t), t] on "Y;.n"Y •.

Differentiating we obtain

L ogj 01P~. + ogj = L Ofjk ogf + Ofjk. ac~

at

ot

azf ot

(2)

at

Then (2) implies [multiplying by (iJlozj)]

~ azj ( a) og~ ~ afjk a ~ ozj ( a ) olP~. ~ ogj a ozf ozj . at + L" at ozj = L" iX~ ozj at + L" at ozj'

L"

3)

(

Hence, '1;..

~ Dg~().)

+ L" ot

[_0_] _~ ag~(\.) [_0_] + 0 IX oz.(;.)

-

L"

ot

II oz'(')

A"

(4)

on "Y,\ n "Y •. Therefore if we let

we get '1,\. - 0;.. = 0. - 0,\.

Q.E.D.

So we see that the infinitesimal deformation, dM ,Idt E H 1(M" 0,} is determined uniquely by the family Jt = {M, It E B} and is thus well defined. If we introduce new coordinates on B, I = t(s) so that t'(s) =1= 0 then the relation (5)

is obvious. Now to return to the more general case, let {M, I t E B} be a family where B is now a general connected complex manifold. Let A be a coordinate neighborhood around bE B and let (t I , ... ,1m) be local coordinates. Then

3.

INFINITESIMAL DEFORMATIONS

39

U}

we may assume II so chosen that w-I(ll) = '¥I j , a union of finitely many coordinate neighborhoods on each of which there are coordinates (zj, "', zj, tl, " ' , t m), where '¥Ij = {(Zj' t)llzjl < Bj' tEll}. Again we have transition functions fjk

zj = fjk(Zk' DEFINITION 3.1. { 0jk I y(t)} where

(oM,/ot Y )

E

tl, " ' ,

HI(M" 0

tm ) on '¥I j n '¥Ik' is the

1)

cohomology

class

of

If (a/at) denotes the tangent vector

a at =

m

a

.f:1 c. ot

Y ,

then we define

We make the following definition: DEFINITION 3.2. .It = {MIl t E B} is locally trivial (complex analytically) if each point bE B has a neighborhood II such that w-I(ll) = Mb x II (complex analytically). This means that we can choose coordinates (zj, t) such that, zj = fMzk' b) (independent of t). If .It is locally trivial, then each M I is complex analytically homeomorphic to M 0; hence M I is independent of t. PROPOSITION 3.2.

If .It is locally trivial then (oM,/ot") = O.

Proo}: Trivial. We mention here a theorem ofW. Fischer and H. Grauert (1965). THEOREM. If each M I is complex analytically homeomorphic to M b , then .It is locally trivial. We now study some examples:

EXAMPLE I. Let R be a compact Riemann surface. Fix a point a E R. Let w be a coordinate in a neighborhood of a point bE R such that w(b) = O. We define a family {M ,} as follows: M, will be the branched two-sheeted

40

SHEAVES AND COHOMOLOGY

covering Rp of R with branch points at a and p; t = w(p). We have the question "is

d:,

= 07" Define the following neighborhoods on R:

Wb = Wo =

WI =

{wllwl < r}, {wllwl < rI2}, {wi rl4 < Iwl < r}.

We can write M, = Uo U UI U U 2 ••• Uj ••• where U o = 1I:- I (WO)' UI = n Wo = 4>for j ~ 0, 1 and 11: is the map 11: : M, -+ R defined by the covering map 11: : Rp -+ R. We introduce local coordinates as follows on M" (I E two): 11: -I( WI), 11:( U j)

Jw - ton Uo , = Fw on U

Zo = ZI

I,

and Zj on Uj can be an arbitrary coordinate which should be fixed and independent of t. Then we have Zj = fjiZk' t) for holomorphic fjk. In fact, Zo = fOI(zl, t) =

Jw - t = Jz~ - t,

and Zj

=

fjiZk) (independent of I)

for (j, k) ¢ {CO, 1), (1,0)}. Then OCt) component,

= {Ojk(t)}

has only one nonzero

(0) = - 2Jz~1 _ t ( OZo0) = - 2zo1 (0) OZo .

ofo I °Ol(t) = Tt· OZo

Let Vo = Uo , VI = UJ ~ I Uj • Then OCt) is a l-cocycIe on the covering "Y = {Vo, Vd; OCt) E HI("Y, e,) s; HI(M" e,). Suppose dM ,Id! = O. Then there are holomorphic vector fields Oy(t) on Vy such that

so (6)

We make the definition

3.

INFINITESIMAL DEFORMA nONS

41

Then ,,(t) is a vector field on M, which is holomorphic on M, - {p} and has a simple pole at p. LEMMA 3.2.

~

If the genus 9 of R is

1, then such vector fields '1 do not exist.

COROLLARY. If 9 ~ 1, then dM,/dt"# 0, that is, the conformal structure of the branched covering M, depends on t.

Proof (of lemma) By the Riemann-Hurwitz formula, we have X(M,) = [2 - 2g(M,)] = 2X(R) - 2 where X(M) is the Euler characteristic of M. Then the genus geM,) equals 2g. By the Riemann-Roch formula [see Hirzebruch (1962)], there is a holomorphicdifferentialqJ(z) =h(z)dzonM,with2(2g) - 2zerossince2g - 1 ~ 1 > O. Since" = y(z)(d/dz) has one (simple) pole, fez) = h{z)y(z) is a meromorphic function on M, with more zeros than poles [2(2g) - 2 ~ 2]. This is impossible Q.E.D. (the number of zeros equals the number of poles). EXAMPLE 2. Ruled Surfaces (See Chapter 1, Sections 3 and 4.) Recall that M, = U'I V U,z where each U'y = C X pi and (ZI' (I) - (Z2, (2)

if and only if '1

We are assuming m

~

= ZiC2 + tz~,

2k, k

~

and

ZI

= Ijz2'

1. Then M, is independent of t "# 0 for t :p 0 so

dMr/dt = 0 for t :p O. (For this, one could use the theorem of Fischer and Grauert.) What is dM ,Idt 1,= 0 ? Consider the covering of M 0, dlJ = {U 01> U02}; then

dM,/dtl,=o = 0(0)

E

Hl(dlJ, 0 0 ) £ Hl(Mo, 0 0 ),

Then

so that adO)

af~2) ,=0 = ( ---at

(0) a'l = k(a) a'i Z2

E

qU ol (', V

02 , 0).

Suppose dM,/dt = 0 at t = O. Then OdO) = ()2 - 01

where each 0. is a holomorphic vector field on UOv = C

X pl.

42

SHEAVES AND COHOMOLOGY

LEMMA 3.3.

Any holomorphic vector field on C x pi is of the form 0= 9(Z)(:z)

+ [a(zK2 + b(zK + C(Z)](:c),

where g, a, b, care holomorphic functions on C. Assume Lemma 3.3. We have the following relations:

(o~J = Z~(a~J. (o~J = mZICI(o~J - zf(o~J,

(7)

where (Zy, Cv) are coordinates on VOy = C X pl. Let us compare the coefficients of (0/oc 1) in O2 - 01 and 012 (0). From Equations (7), we get Z~

I

= zic 2(Z2) - C (Zl) = zic 2 (Z2) -

CI

CIJ,

where the cy(z.,) are entire functions. Expanding,

and 0 < k < m. This is impossible. Hence,

dMtl /1=0 =F O. dt For the lemma we have: Proof. Let (z, 0 E C X pi, where Cis a nonhomogeneous coordinate on At C= 00., the local coordinate on pi is 1'/ = llC. Restrict the vector field to C x pi - C X {oo} = C 2 • Here pl.

0= 9(Z,C)(:z) + h(Z,C)(:C)' where 9 and hare holomorphic on C 2 • At O=y(z, I'/)(:z)

where C=

00

we have

+ P(Z,I'/)(:I'/)'

1/'1 and y, pare holomorphic. Then oC/ol'/ = -1/1'/2 so (0/01'/)

- C2 (0/00. Hence at

00,

0= y(z,

I'/)(:J -

,2{3(Z,

'1)(:,),

=

3.

INFINITESIMAL DEFORMATIONS

43

since g(z, ') = y(z, "'), g(z, ~) is holomorphic on C x pl. So g(z, ') is constant as a function of ,

g(z,

0

= g(z).

Finally, h(z, 0 = - ,2 P(z, "') implies that h(z, ') has a pole of order::;; 2 at So h(z, ') = a(zK2 + b(zK + c(z). Q.E.D.

00.

REMARK 1. The dime HO[M(m), 0] is the number of (complex) linearly independent holomorphic vector fields on M(m). We want to compute it. As usual, M(m) = UI U U2 , U. = C X pi, and

if and only if

We must count the number of parameters involved in representing a HO[M(m), 0]. By the lemma,

() E

() = (). = g.(Z')(a~v) + a.(z.K~ + bv(z.K. + c.(Zv)(a~J on each U., and

(}I

=

(}2

on UI n U2. Changing coordinates,

Hence

(}2 = -Zig2(a~J + mZlg2'I(a~J + (a2 zim'i + b2 z~' I + C2)Z2(a~

J

=

-Zig2(a~J + [a2 z~'i + (b 2 + mz l g 2)'1 + C2 zim](a~J

=

(}I = gl (a~J + (alCi + b l ,! + Ct)(a~J·

Equating coefficients,

gl(Zt) = -zIgiz2), a1(zt) = zTaiz2)' b1(z.) = b2(Z2)

+ mz.g 2(z2), C!(Zl) = Zi'"C2(Z2).

44

SHEAVES AND COHOMOLOGY

These functions are all entire functions of Zl' Let us investigate their behavior at ZI = 00. Since Z2 = lIz!> we see that 91 has a pole of order ~2 at 00, al has a pole of order ~m at 00 and CI has a zero at 00. Assume that m ~ 1. Then

z~

gl

=

al

= alOz~

CI

= 0

glO

+ gl1 Z 1 + g12' + ... + aim,

(by Liouville's theorem).

Consider the b terms:

So bl(zl) = -mg lo zl blO . 0 depends linearly on (910,911,912' alO' "', aim, blo), Hence,

(8) We therefore have: THEOREM 3.1.

M(m)

=1=

=1=

M(n)

REMARK 2.

M(2n)

REMARK 3.

Let {M tit E

(complex analytically) if n =1= m.

M(2n-l)

topol09 ically.

q, M t given y

':01

=

my Z2':o2

as before. Then Mo = Ml m), M, =

dM,ldt={

by

+ tz k2 , ZI M(m-2k)

0, =1=0,

1

=Z2

for t =1= O. And we have shown

fort=l=O

for t = O.

Suppose we "reparametrize" and consider {M.21 sEq.

Then Mo

= M(m),

Ms2

= M(m-2k),

S

M.2

is defined by

=1= 0 as above. But

dMs2 = dM t • ds 2 = 2s dM, = 0 ds dt ds dt for all SEC. We know that M t independent of t implies dMt/dt = O. We have just seen that dM,/dt = 0 does not imply that M, is independent of t. However, we have the following theorems:

3.

INFINITESIMAL DEFORMAnONS

4S

THEOREM Ol. If dim HI(M I> 0,) is independent of I and if iJM ,liJI' = 0 for all v and I, then {M, I E B} is locally trivial and hence M, is independent of I. THEOREM p. The function t function of I. That is

-+

H1(M" 0,) is an upper semicontinuous

dim HI(M" 0,) ~ HI(Ms' 0 s), if I is in a sufficiently small neighborhood of s; that is, lim dim H1(M" 0,) ~ Hl(Ms. 0 s). , .... s

THEOREM y. of s.

If Hl(Ms' Os) = 0, then M, = Ms for t in a small neighborhood

Theorem Ol is proved in Kodaira and Spencer (l958a), Theorem p in Kodaira and Spencer (1960), and Theorem y is due to Frolicher and Nijenhuis (1951). Theorem p follows from some results which we will prove in a later chapter. Theorem ex will not be proved here. DEFINITION 3.3. We say that a compact complex manifold M is rigid if, for any complex analytic family {M, It E B} such that M,o = M, we can find a neighborhood N of to such that M, = M,o for lEN. (More precisely, if w: J( -+ B is the family {M,}, then w-I(N) = N x M,o complex analytically.) The following theorem follows from Theorem "/. THEOREM 3.2. If Hl(M, 0) = 0, then M is rigid. We will give a proof of this using elementary methods. We have the following: PROBLEM. (Not easy?)

Find an example of an M which is rigid, but Hl(M, 0) '=F O.

REMARK. IfDn is rigid. For n ~ 2 the only known proof is to show H 1(lfD n, 0) = 0 [Bott (1957)]. Let us proceed to the proof. Proof (of Theorem 3.2) The proof will be elementary in that it consists of two elementary ideas:

(I) (2)

Construction of a formal power series, and proof of convergence.

The proof is actually long and computational, so please stay with us. It makes no difference for the proof and it makes the writing much easier if we assume

46

SHEAVES AND COHOMOLOGY

dim B = 1. The result is local so we may assume B = {tlltl < r} and to = O. We can cover (jj-l(d£), d a = {tlltl < e} with coordinates o//j = {(Zj' t)llzjl < ej' It I < e}. Then

Zj=fjk(Zk,t)

on

o//jno//k'

M is covered by u UJ = M where j

UJ Then M x B

= u(UJ

= {z j Ilzjl < l)X{O}

x B) where for (wv, t)e

!:

U~

o//j.

x B,

J

if and only if

that is, (9)

wj = gMwk), We can rephrase our result:

If b is sufficiently small there is a biholomorphic map qJ of (jj-l(d,) onto M x d, such that qJ: maps (jj-l(t) onto M x t and qJ: M = (jj-l(O) -+ M x 0 is the identity map. Suppose we choose b so that qJ maps

THEOREM.

into U~ x B, qJ(o//i)!: U~ x B. Let (zj,t)e0//1. Then, qJ(Zj,t) =(wj,t) = [qJj(Zj, t), t] so on each qJ is represented by holomorphic functions qJ~(Zj, t) where qJj(Zj' 0) = zj. On o//~ n

0//1,

all:,

zj = fMzk' t); so

implies (10)

Therefore we see that we can prove the theorem if we can construct holomorphic functions qJj(zJ' t) on o//~ satisfying (10) and

qJj(Zj' 0)

= zj.

(11)

3.

INFINITESIMAL DEFORMATIONS

47

For simplicity we may as well assume that dIl j is of the form

= {(Zj' t)llzjl < 1, It I < e}, M = U UJ, {zjllzjl < 1 + v for some v > O} and UJ:::> M n dIl j • If expand CPj(Zj, t) dIl j

UJ = into a power series, we get CPj(Zj, t)

= Zj + CPjI1(Zj)t + CPjI2(Zj)t 2 + '" + CPjlm(Zj)t m + ... ,

(12)

where each CPj/m(z) is a holomorphic vector valued function. If we expand both sides of (10) we get co

co

L Fm(CPjll' ... , CPjlm) = m=O L Gm(CPt/1' ... , CPltlm)tm, m=O

(13)

where Fm and Gm are polynomials. We introduce some notation: If P(t) = Pn t" and Q(t) = Qn t n are two power series, P(t) == Q(t) means Qn = P n

L

L

m

un to n = m [that is, P(t) == Q(t) mod (tm+l)]. Therefore, to solve (formally) Equation (13) we need only solve CPj[fjk(Zk, t), t] == gjk[CPk'(Zk' 0], m

for each m, where cPj(Zj' t) =

Zj

+ ... + CPj/m(z)t m.

(14)

First consider m = 1. We have co

Zj

= fjk(Zk, t) = Yj/.(zk)

+ L fjklm(Zk)t m • m=1

Using (13)10 gjk(Zk)

+ fjkll(Zk)t + CPjJ1[gjk(Zk)]t == gjk[Zk + CPkl1(Zk)t] 1

So

Now

ejk = ~ (a~:k),=oC~~j) = L fjkll(a~j) belongs to Hl(M, 0). By assumption HI(M, 0) = 0 so {Ojk} is cohomologous to zero. But Equation (13)1 says we must find {CPk)!} so that Ojk = CPk/1
48

SHEAVES AND COHOMOLOGY

Assume that ({)j(Zj' t) are determined so that Equation (13)m holds, that == r jk t m +I. We must show that ({) jim +1 (Zj) can be is, ({))"!(jjJ - 9 jk( ql:) m+l determined so that ({)j+l

= ({)j + ({)jjm+lt m+1 satisfies (13)m+1> which is

({)j+l[fjk(Zk, t), t] - gjk«({)r+ 1) == O.

m+l

This is equivalent to ({)j(fjk)

+ ({)jlm+l[fjk(zk, t)]tm+1 ==

m+1

gjk[({)r(Zk, t)

+ ({)klm+l(Zk)' tm+l].

We use

Here we have used

and ({)r(Zk, t)

= Zk

+ .. '. So if we can solve rjk(Z) =

L/1 uZk ~z~ ({)klm+l(zk) -

(15)

({)jlm+l(Zj),

for ({)Jlm+1 we will have (l3)m+I' Let r jk =

({)jlrn+ 1 =

~ rjk(Z)(O~j)' ~ ({)j\m+I(Z)C7~j)'

Then we want to solve r jk = ({)k \ m+I - ({) j \ m+I' We claim that {r jk} is a cocycle [belongs to Ht(M, 0)]. Then we would be done as before, since Ht(M, 0) = 0 and {rjd must be a coboundary, r jk = Ok - OJ, and set OJ\m+l = OJ. LEMMA 3.4. That is,

{rjdisacocycle,thatis,rik = r ij

+ rjkon"lli /l o/i j

/l

"lI k

/l

M.

3. Proof

49

INFINITESIMAL DEFORMATIONS

By definition

rik(Zj)t m+ 1 == CPi'[fik(Zk, t), t] - glk[cpr(Zk, t)], m+l

so

Now

r Jk t m+ 1 == CPj(fjk) -

9 jk( cpr) and

m+l

So

By assumption

CPi'[fij(Zj, t), t] - Yij[cpj(Zj, t)] == rij(Zj)t m + 1, m+l

== rij[fjk(Zk, t)]tm+ 1 -

••••

m+l

Hence

Q.E.D. This finishes the construction of the formal power series

cp}Zj' t) = Zj Zj E dlt j

+ CPjll(Zj)t + ... ,

n M, dlt j n M = {zjllzjlll < I}

such that CPJjjk(Zk' t)] = 9jk[CPk(Zk' t)J as formal power series. LEMMA 3.5.

() >

o.

The power series CPj(Zj, t) converges for It I < () for some small

Proof We dominate Cpj with a convergent series. We fix some notation. Let t/I(z, t) = 'L:=ot/lm(z)t m be a power series where t/lm(z) =[t/I~,(Z), ... , t/I~,(Z)J zE U. Let a(t) = 'L:=o amt m , am ~ 0 be a series with real, positive coefficients. We write t/I(z, t) ~ a(t) and say that aCt) dominates t/I(z, t) if

50

SHEAYES AND COHOMOLOGY

I",=.(Z) I ~ am for all Z E U and all ex max sup 1",=.(z)l. Consider the series

= 1,

... , n. The norm of "''" is

I"'ml =

zeU

II

b 00 (et)m A(t)=- L - 2 ' 16e m= 1 m

where band e are constants to be determined later. Then b {

A(t) = 16 t converges for

It I <

00

em - l t m }

+ m"'f2 -;;:;r

lIe. In Lemma 3.5 it suffices to prove l{)j(Zj, t) - Zj ~ A(t).

(16)

In fact, it suffices to prove t) -

l{)j(Zj,

Zj

~

for m = 1,2,3, ....

A(t)

First, we prove: LEMMA

3.6.

[A(t)]2 ~ (ble)A(t).

[A(t)]2 = (l!...)2[ f (et~m] [f

Proof·

m=l m

16e

= (-

b

)2 L(et)" 00

16e

n=2

k=l

(ett] k

1 L 2"2' m+k=n m k

Since 1

1

L 2"2~2k+m=n L 2"2' m+k=n m k m k msk

and m

~

k implies k

~

n12, we get

1

2

00

mh=n m2k 2~ (n/2)2 m"'f

1 1 m2

8 n2

16

~ n2"6 < n2.

Hence,

(b)2 L~ -(et)" 6 b b ~ (et)" - 2 1 ~--L--2'

[A( t)] 2 ~ -

16e

COROLLARY.

n= 2

n

e 16e n = 1 n

A"(t) ~ (b/e)n-l A(t).

Let us prove (16)1: We want to show that l{)i11(Zj)t ~

b A(t) = 16 {t

+ .. '}.

Q.E.D.

3.

INFINITESIMAL DEFORMAnONS

51

It is enough to prove \qJj 11(Z) \ ~b/16. From (13)1>

L ~z~ qJfll(Zk) -

qJjll(Zj) = fj"ll(Zk)'

UZk

We may (perhaps by shrinking Um to U! so that u V! = M) assume that the given functions jj" 11 are bounded. Also the qJ j 11 (z j) are holomorphic on and we may, therefore, assume a finite upper bound for all of them. (Compactness is needed for these statements.) So we see (16)1 is satisfied for b large enough. By induction, assume (16)m and let us prove (16)m+l' Remember

V:

qJjm+l

= qJjm + qJjJm+l (Zj )tm+1 ,

and

r j,,(Zk)tm+1 ==

m+l

qJjUjk(Zk, t), t] - gjk[qJ~(Zk' t)].

We fix some more notation: For 1/I(t) =

L 1/Im t m, let

[I/I(t)]m+l = I/Im+ltm+l.

Then

Remember the definitions of U j and UJ, U j C UJ. Then gjk(Zk) is defined on UJ n U~ and gjk(Zk + Y), Y = (Yl, "', Yn) is holomorphic for Zk E U j n U", \Y\ < v. So there is a " >0 so that 00

gjk(Z"

+ y) ~

L "m(YI + ... + Yn)m. m=O

(17)

For the moment, let 1/I"(z,,, t) = qJ~(Zk' t) - Zk' We want to estimate [gj,,(qJ~)]m+l = [gjk(Z" + 1/IJJ+ml where m + 1 ~ 2. But,

since I/I"(z,,, t) is a polynomial of degree we get

~

m in t. From Equation (17),

52

SHEAVES AND COHOMOLOGY

hence, 00

[gjk(Zk

+ !/Ik)]m+ 1 ~

L KTnA(t)J ,=2

~ A(t)

L K'n' (b)'-1 C 00

,=2 K2

1

n2 b

= A{t)-c-1 _ (Knh/c) ,

because by induction!/lk

~

A(t).lfwe choose c so large that (Knb/c) <

{gjk[tp~(Zk'

t)]}m+l ~

2K 2 n 2 b c

A(t).

1, then (18)

Now we want to estimate rjk(Zj)t m + 1

== tpj[Jjk(Zk, t), t] - gjk[tpk'(Zk, t)J. m+l

Remember that

r

is a cocycIe so

azrp r ljkX on Ui n r iklX = rlXij + "~ -a Zj

Uj n Uk'

where U i satisfies U i = {zillzil < I}. We choose ut c Ui> Ui = {zillzil < 1 - P} such that u Ui = M. It suffices to estimate rik(Zi) for Z E ut n U i , for take any Z E Uj n Uk, then Z E ui for some i, and

r~k(Zj) = L aaz~ {rMzj) - niz;)}. Zi IX

Then if r jk(Zj)t m+ 1 ~ B(t) for Z E Uj n Uk, where

K 1 = 2n

Z E

Uj n Uk>

r jk(Zj)t m+ 1 ~ K 1B(t) for

n~3:x (sup IaZi az~ \). I,]

So consider tpj[fjk(Zk, t), t] for

Uj n Uk' Expanding./jk'

ZE

!jk(Zk, t) =

Zj

+ !jkll(Zk)t +"',

we see we may assume that

for small t and some constants bo , tpj -

Zj

=

Co •

Our induction hypothesis is

m

00

,,=1

1

L tpjJ,.(z)t" ~ A(t) = L a"t".

3.

53

INFINITESIMAL DEFORMATIONS

Since each lfJ j I", is holomorphic, we get a power series

lfJjl".(Zj + Y) -lfJ.iI".(z) =

L

VI+"'+v"2:1

cVI'''Vny~1 ... Y~",

where

Since IlfJn ",I < a", for some constant a"" on IYIII =

Ic

" •

••• \In

/3,

we see that

I ~ (a Il I/3VI ... Vn).

We thus obtain

Summing we get (19) Again, since lfJj(Zj' t) only has terms of degree [lfJj[fjk(Zk, t), t]]III+l

=

~

m in till we obtain

[lfJj(Zj' (1jk, t) -lfJj(Zj' t)]m+l·

Using (19) we get . + Vn

The corollary to Lemma 3.6 gives Ao(t)" ~ (b olco)v-l Ao(t), so

[ f= v

0

(Ao)v]n -1

/3

~

[1 + f= v

=

1

~(bo)V-1Ao(t)]n_1 /3 Co

[1 +~ _(b~//3co)Ao(t)r-1. 1

Since we have chosen Co so that Ao(t) ~ fjk(Zk, t) - Zk' we may replace Co with a larger constant and assume that bo/2c o < t. Then

54

SHEAVES AND COHOMOLOGY

where K2 is a constant depending only on n,

[CPj[fjk(Zk' t), t]]m+ 1

p, bo , and co. Thus,

~ ~2 A(t)Ao(t).

We may assume b > bo , c> Co and then A(t)

~

Ao(t), so

K2 K2b 7i Ao(t)A(t) ~ Ii ~ A(t). Finally,

If - 2K 2n 2) ~b A(t),

K2 ~ ( for

ZE

U; n Uk. Hence

r.

Jk

K b 2 n2 ) -A(t) t m + 1 ~K ( -2+2K 1

(3

(20)

c'

~ ozj p () r ajk = L.. :;p CPklm+ 1 Zk vZk

a

(_ )

qJjlm+ 1 "" j

and we want to estimate cpjlm+l(Z). As before, consider r jk = L rjizj)(%zj), CPjlm+l = L cpjlm+l(O/Ozj). With these notations, r jk = CPklm+l - CPjlm+l. At this point we need a lemma which plays a crucial role in arguments of this type. Let r = {r jk } be a l-cocyc1e where r jk is a holomorphic vector field on Uj n Uk. Let '" = {'"j} be an O-cochain where '" j is a holomorphic vector field on U j • We define

lin = max j,k

sup

max

ZeVj"Vk

a

WMz)l.

""'II = max sup max l"'j(z)l. j

ZE

VJ

a

LEMMA 3.7. There exists a constant K such that if r is cohomologous to 0, then we can find", with fJ", = r satisfying" "''' ~ K where M is a compact complex manifold, Hl(M, 0) need not be zero, and K does not depend on r.

"n

Proof Remember Vj = {zjllzjl < I} and U;{zjllzjl < I the lemma is not true. Let t(r) = {""''' I fJ", = r} <

Pl. Suppose

+ 00.

I)"n.

Ift(r) ~ K"n, then there is '" such thatfJ", = rand ""''' ~(K + Thus if the lemma is false, term is unbounded. Hence there is a sequence {reV'}

n

3.

INFINITESIMAL DEFORMATIONS

55

such that r[r(·)]/lIr(·)II-+ 00 as v -+ 00. Replace r(·) with r(·)/r[r(·)]. Then r[r(·)] = 1 and IIf(·)II-+ O. Hence we can find {I/I~.)} holomorphic on Uk so that 1II/I(v)1I < 2 and q:) = (){I/I1 V )} = I/I1 V) - I/I~V). By Vitali's theorem we can select a subsequence {I/Ir>.)} such that I/I~V,l)(Zj) converges uniformly on Ur If Z E Uk then Z is in some Ur Hence for each k, 1/11·,l)(z) = I/I}·,l)(z) + r~~,l)(z) converges uniformly on Uk so cp~·,l)(z) -+ I/Ik(Z) uniformly on Uk. Since f~~,l) -+ 0, I/Iiz) = I/Ik(Z) on Uj n Uk. Let tfr)V,l) = I/Ii·,l) - I/Ij. Then q:,l) = tfr~·,l) - tfr}V,l). But IItfrc· . . )1I < 1- for large v, and thus r[r c.... )] < t. This contradiction proves the lemma. Q.E.D. By Lemma 3.7 we can find K and {CPjlm+l} so that r jk = CPklm+l - CPjlm+l and IICPm+ll1 ~ Kllfll· Hence

(2 2+73 K2) ~A(t). b

CPjlm+l(Z)t m+l ~KKI 2K n

We can choose c so large that KKl(2K 2 n 2 + K2/P)(b/c) < l. Then CPjlm+1 t m+ 1 ~ A(t) so cpj+l(Zj' t) - Zj ~ A(t). This completes the induction and proves that cP j(Zj, t) - Zj converges for small t, thus finishing the proof of the theorem. Q.E.D. There is the following: THEOREM. [Kodaira, Nirenberg, and Spencer, (1958)] Assume H2(M, 8) = 0 (M is compact complex as always). Then for any element e E Hl(M, 8), there is a complex analytic family {M,lltl < r, r > O}, such that Mo = M and (dM,/dt),;o = e. PROBLEM. Find an elementary proof of this. (In the analogous idea of proof the convergence gives trouble.) We also have the completeness theorems. DEFINITION 3.4. We say that the family (.it, B, w) is complete at be B if, for any family (.AI", A, n) such that n- 1 (a) = w- 1 (b) = Mb there is a neighborhood U 3 a and holomorphic maps <1>: n- 1(U) -+.it, h: U -+ B such that h(a) = b, 7[-l(V) _ _ .it

·1 . 1

commutes,

0

U--B

maps 7[-l(S) biholomorphically onto w-1[h(s)] for each s E U, and : n-I(a) = Mb -+ Mb is the identity map.

SHEAVES AND COHOMOLOGY

56

Roughly, (...II, B, w) is complete if it contains all sufficiently small deformations of M b. The (holomorphic) tangent space Tb at b is the set

{:t \:t vt/v(o~v)}. =

map Pb: (a/at) -+ (OMrlOt)r=b Tb -+ Hl(Mb' 0 b).

The

THEOREM.

[of completeness; see

E

Hl(Mb' 0 b) defines a linear map

Kodaira and Spencer (1958b)]

If

Pb(Tb) = Hl(Mb' 0 b), then (...II, B, w) is complete at b. REMARKS. (1) Theorem 3.2 is a corollary of this theorem. (2) This theorem of completeness can be proved by the same elementary method used to prove Theorem 3.2.

4.

Exact Sequences As usual X is a paracompact Hausdorff space, [/ is a sheaf over X and map.

w : [/ -+ X is its projection DEFINITION 4.1. (1) (2)

(3)

[/'!; [/ is a subsheaf if

[/' is open, w([/') = X, w-l(x) n [/' = [/~ is an R-submodule of [/".

Let [/" be a sheaf over X with projection

w".

DEFINITION 4.2. A homomorphism h of [/ into [/" is a continuous map of [/ into [/" such that (1) (2)

w"

0

h = W,

h: [/" -+ [/; is an R-homomorphism. h is a local homeomorphism. We define the kernel of h to be

REMARK.

ker h = {s I h (s) = O} where" t = 0" means t" = Ox E [/x. LEMMA 4.1.

ker h is a subsheaf of [/.

LEMMA 4.2. h([/) is a subsheaf of [/". The proofs are left to the reader.

4.

57

EXACT SEQUENCES

Let f/' E f/ be a subsheaf. Let Qx = f/x/f/~ which is an R-module. Define hx: f/ x -+ Qx to be the natural homomorphism. Let Q = UXEX Qx' Define 11: on Q by 1I:(Qx) = x; h is defined h : f/ -+ Q by h(s) = hx(s) for s E Y X' We give Q a topology by saying i1IJ is open if and only if h -1(011) is open in f/ (the quotient topology).

LEMMA 4.3. Proof

Q is a sheaf and h : f/

-+

Q is a surjective homomorphism.

Left to the reader.

DEFINITION 4.3.

Q is the quotient shea/of Y by Y' and we write Q =Y/f/'.

A sequence ho

III

f/ 0 - - + f/ 1 - - + f/ 2 - - +

... - - +

IJ ,.+

hn

J

Y n - - + Y n + 1 - - - + ...

of sheaves is exact if h.(f/.) = ker(h.+ 1) for all v. Suppose given sheaves Y and Y" over X and a homomorphism h : Y -+ Y". Let V be an open subset of X and Y) be the set of all sections of Y over V, Then h 0 U E Y") if U E Y) so h induces a map h: Y) -+ V, f/"). Let i1IJ = {Vi} be a locally finite covering and Cq(OlI, Y) be the space of q-cochains cq = {u io ... i.} where U io ... j. E V jo n ... n V j . ' Y). Then h induces a map h : CQ(i1IJ, f/) -+ Cq(i1IJ, f/") defined by h cq = {hUio'" i.}' Then we have:

re

LEMMA 4.4. Proof

reV, reV,

reV,

reV,

re

h0

()

= () h. 0

Obvious.

Hence h maps zq(OU, Y) into zq(OU, f/") and thus h induces a homomorphism h: Hq( i1IJ, f/) -+ Hq( i1IJ, f/"). Let 1f/ = {1f/./.} be a refinement of i1IJ, 1f/ > i1IJ. Then HCJ(u71, f/)~Hq(i1IJ, f/")

ln~

In~r

Hq( "If", Y)-..-!:.- Hq( 1f/ f/")

commutes. Hence h induces a homomorphism h: Hq(X, f/)

-+

Hq(X, Y").

SHEA YES AND COHOMOLOGY

58 THEOREM 4.1.

Assume that

is exact. Then there is a homomorphism c5* such that

is exact.

Proof i is injective so f/' ~ i(f/') c f/, and i(f/') = ker h. Thus we consider f/' c f/ where f/' = ker hand i is the inclusion map. Recall that HO(X, f/) = ZO(X, f/) = f/). Since nX, f/') c nX, f/), we see that 0-+ HO(X, f/') ~ HO(X, f/) is exact. If a E nX, f/), then ha = 0 if and only if a E nX, f/'); so HO(X, f/') ~ HO(X, f/)..!!. HO(X, f/") is exact.

rex,

LEMMA 4.5. HO(X, f/)..!!. HO(X, f/")~ Hl(X, f/') is exact (where we must define c5*).

Proof Let a" E nX, f/"). Since h is a local homomorphism there is a section Ty E Uy , f/) over a small neighborhood Uy of y such that h Ty(X) = a"(x) for x E Uy . Now {Uy lYE X} covers X and we have a locally finite refinement tft = {Uj } of {Uy}, that is, there is a map j -+ y(j) such that Uj S;; Uy(j)' Set Tj = r Uj ry(j) E nUj' f/). Then h Tj = a" where defined. Let CO = {Tj } E CO (tft, f/). Then:

n

DEFINITION (4.4)1' c5*a" = [c5eO] E Hl(X, f/') where for any eq E zq(tft, f/), [e q] denotes the cohomology class in Hq(X, f/) of eq. (One should check that c5* is well defined.) Since c5eo = {Tk - Tj} and hTk - hTj = a" - a" = 0, we see that c5eo E ZI(tft, f/') so Definition (4.4)1 makes sense. Exactness means c5*a" = 0 ifand only if a" = ha for some a E nX, f/). So suppose (j*a" = [c5eO] = O. Then c5eo = (jeo' where co' = {Tj} E eO(tft, f/'). So CO - co' = a E ZO(X, f/) = nX f/), and ha = heo = {hT j } = a". Now suppose a" = ha. Then h(Tj - a) = 0, so Tj-aEnUj,f/'). Set C~={Tj-a}=co-aECO(tft,f/'). Then c5e~ = c5eo since a E ZO(X, f/) and hence c5*a" = [c5eo] = [c5e~] = O. Q.E.D. We now turn to check that

4.

59

EXACT SEQUENCES

is exact. Take e l ' E ZI(Olt, f/'). If [e l '] = {)*q" = [{)eO], then i[e l '] = 0, and if ;[e l '] = 0, then e l ' = {)eO; so 0 = he l ' = Meo = ()heo. Thus heo defines an element q" E reX, f/"). By definition, [e l '] = {)*q". We want to prove exactness

LEMMA 4.6. Given ell" E Cq(qj,f/"), then we can find a locally finite refinement "if'" and ell E CIl("if'", 9') so that n~eq" = heq.

Proof We give proof for q = 2. Let Olt = {Vi}' eq" = {qijd, where q/jk E r( Vi (1 Vi (1 V k , f/"). Choose a covering "Y = {Vi} such that Vi C V J • Since Olt is locally finite, a given Y E X belongs to only finitely many Vi' We choose a neighborhood Ny of Y sufficiently small so that x

E

(I) if Y E Uk (1 Ui (1 V k there is -r E r(Ny , f/) with q/jk(X) = h-r(x) for Ny (remember h is a local homeomorphism), (2) for each y there is Vi such that Ny C Vi' (3) if Ny (1 tj :F cP then Ny C Ui .

Then {Ny lYE X} covers X and we can choose "if'" = {WJ a locally finite refinement of {Ny}. Hence there is a map A. - YA such that WA C Ny). • By (2) Ny). C Vi;>.' so 1(/ > "Y > Olt. Define -r = {-r A".} E C 2 (1(/, f/) as follows: we have WAC VI' W"c Vi' WycCk where i=jA' j=jA, k=jy. By (3) if Ny). n Vi :F cP, Ny). C Ui , and Ny). n Vk:F cP gives NYJ.. C Ub and so on. We are assuming WA n W" n W. :F cP, and WA C Ny). C Vj , and so on. Hence it follows that YA E Ny). C V k (1 Ui n Uk' By (1) q/jk(X) = h-r(x) for x E Ny). where L E r(Ny". f/). Let LA". = rWA" W .... " Wy(L). Then

Q.E.D. Let us prove that h

j

Hq(X, f/)-Hq(X, f/)-Hq(f/")

is exact. hi = 0 is clear. Suppose,., E BIl(X, f/) and h,., = O. Then,., = [eq], eq E ZIl(Olt, f/) for some Olt and n!!-hell = ()e ll - I " for some "Y and eq- I " E Cq-l("Y, f/"). By the lemma n~eq-I" = h Iq-I for some 1(/ and E cq-I(1(/, f/). Thus

,,-1

hn~ ell - h {)t,-I =

0,

so n~ eq - ()t q- I = eq'

where eql E zq(1(/, f/') ~ cq(1(/, f/'). Finally we get ,., " = i,,' where ,.,' = [eq,] E Bq(X, f/').

= [eq] = [e"]

so

60

SHEAVES AND COHOMOLOGY Next we prove that

is exact. We must define 15*. Take '1" E Hq(X, f/"). Then '1" = [eq"], eq" E zq(O/t, f/"). By the lemma there is t q E Hq(1I', f/) such that h t q = IT:' e q". DEFINITION 4.4. 15*'1" = [t5tq] E Hq+l(X, 9"). Again we should check that 15* is well defined. For the moment denote n:. by n. Suppose 15*'1 = O. Then TIt5t q = bbq' for some bq' E C q(1Ji, 9"). Thus t5(IT t q - bq,) = 0 and n t q - bq = eq E zq(1I', 9'). So let ,,= Ceq] E Hq(X ,9'). Then h'1 = [he q] = [TI ht q] = [IT eq,,] = ,,". Suppose conversely that '1" = h". Let" = Ceq]. Then nil = [he q] and ht' = IT heq by definition of t q. So t q - neq = aq, E Cq(11', f/') and (j*'1" = [bt q ] = [t5a q ,]. Thus (j*'1" = O. Finally we prove that 6'

i

Hq(x, f/")-..Hq+l(X, f/')-..Hq+l(X, f/)

is exact. Certainly i (j*,," = O. Since by definition (j*'1 = [(jt q] where t q E Cq (11', f/) so i(j*" = O. Suppose i'1' = 0, ,,' = [e q+1 '] E Hq+l(X, 9"). Then TIe'+!' = (jt q for t q E Cq(1I', f/). Then 0 = (j ht q E 2'(11',9''') and ,," = [htq] E Hq(X, f/"). Since (j*,," = [(jt q] = '1' we are finished. Q.E.D. [If the reader wishes more details for these elementary properties of sheaves he may consult Hirzebruch (1962).] Next we prove functoriality. THEOREM 4.2.

If i

h

0 - . . f/' - . . f / - . . f/"-..O

j~

j~' t

j~. k

O-..ff'-..ff-..ff"-..O

is exact and commutative, then

is exact and commutative.

4. Proof.

61

EXACT SEQUENCES

We need only prove commutativity. We check that Hq(X, !I''') -

6'

Hq+ I(X, !I")

\..

\.- ."

Hq(X, ffll) -

Hq+ I(X, ff')

commutes. The rest is easy. Let,," E Hq(X, !I'''). Then there is cq" E zq(1I/, !I''') and t 4 E C4(1r,!I') such that ,," = [c q"] and cq" = ht q. Thus ~*17" = [c5tq]. However, cp'~*,," = [cp'~tq] = [~cptq], and cp",," = [cp"htq] = [cp"ht q] = [h cpt q ]. Thus ~*cp",," = [c5 cpt q] = cp'c5*,,". Q.E.D. We give a brief discussion of fine sheaves. DEFINITION 4.5. !I' is afine sheaf if for any locally finite covering {Uj} of X there exists a set {hj} of homomorphisms hj: 9.-.9 such that (1) (2)

hj!l'" = 0 for x ¢ Wj' where Wj j = id.

Lh

£;

Uj is a closed subset of Uj,

j

EXAMPLE. Let!l} be the sheaf of germs of differentiable functons on a differentiable manifold X. We have a partition of unity subordinate to Uj ; that is, a set {Pj} of differentiable functions Pj = pix) on X such that (1) (2)

Pj(x) = 0 for x ¢ Wj'

L Pj = 1.

For any local differentiable function f Then h j induces a homomorphism hj : P} is fine. THEOREM 4.3.

= f(x)

on X, define h jf = pix) f(x). !l} -+!l}. Using these {hJ we see that

If 9 is a fine, then Hq(X, 9) = 0 for q

~

1.

Proof. We give the proof for the case q = 2. Let c 2 be a cocycle, = {O'ijk} E Z2(atJ, !I'), with O'ijk E n Uj n Uk' 9). By fine-ness we have the {hj} in Definition 4.5. Since ~C2 = 0, if Ui n Uj n Uk n Ut =F tP then O'jkl - O'iM + O'ijl - O'ijk = o. Since hiO'ij/,(x) = 0 for x ¢ Wi' hiO'ijk can be extended to T ijk E U j n Uk,!I') by setting T ijk(X) = 0 for x E Uj n Uk - Wi. For fixed (j, k) we have only a finite number of Ui with Ui n Uj n Uk =F cp and for each i we have Tijk from hiO'ijk. We set Tjk = Li Tijk. Then

reUi

c2

re

hi O'jkl = hi O'ikl - hi O'jjt + hi O'jjk· Thus O'jkt = Tkl - Tjt proof for any q ~ 1.

+ Tjk so c 2 = &1 Q.E.D.

where c l

= {Tjd. It is easy to give this

62

SHEAVES AND COHOMOLOGY

5.

Vector Bundles

We give a brief review of vector bundles. Again, a good reference for this section is Hirzebruch (1962). Let M be a complex (differentiable) manifold.

5.1. By a complex analytic (differentiable) vector bundle (e" or bundle) we mean a complex (differentiable) manifold F together with a holomorphic (differentiable) map rc: F -+ M onto M such that, for a sufficiently fine locally finite covering dIJ = {V j} of M: DEFINITION ~"

(1) There is an analytic (differentiable) equivalence Ij between rc- 1(V j ) and Vj x en (or U j x ~II) such that

rc- 1(V j )-.!!.- Vj x

en

·1 " 1·, Vj

commutes, where rciZj' () (2) (or Vj

If(z, X ~"),

=

---+

Vj

Zj'

(J. ... , (j) E Vj x e" (or Vj x ~") and (z, (L"', me V

k

x

e"

then ~I f j Jk- 1( Z, ~k' 0

YII)" = l.J"

••• ,C,k

fIXjkP (Z )~k' ~P

P:l

are holomorphic (differentiable) functions on Vj n V k • In vector notation.ljk is the matrix (.Ij~p) and (i = (j), and then

where.lj~p(z)

('J ' .. "

We call rc- 1 (z) the fibre of F over. By (1) and (2) we can give it a vector space structure rc- 1(z) = z X e" [or rc- 1(z) = Z X ~"]. DEFINITION 5.2. We say that Fand F' are holomorphically (or differentiably) equivalent if there is a biholomorphic (bidifferentiable) map cp : F -+ F' such that

(I)

F~F'

\/ M

commutes.

5.

63

VECTOR BUNDLES

(2) On each fibre cp is a linear transformation, that is, if Uj is chosen as in Definition 5.1, then there is a holomorphic (differentiable) matrix-valued function h j on U j such that fi 0 cp Ofj-I = hj, that is, C,/ = hjp(z) C~. In particular, if Fis a C l bundle and Fis trivial over Uj (that is, n-I(U j ) = U j x C), then z E Uj n Uk implies that (z, C) is identified with (z, (k) if and only if (j = fjk(zKk wherefjiz) is a nonvanishing holomorphic (differentiable function) of Z E Uj n Uk. Let (J* (or .@*) be the sheaf over M of non vanishing holomorphic (or differentiable) functions in which the module operation on each stalk is multiplication and the ring R = 7L so here instead of a.f(z) + f3 g(z) we have [f(Z)JIZ[g(Z)]P, a., f3 E 7L. Consider fjk as an element of

Lp

Hence {fjk} E Zl(au, (J* (or .@*» and two bundles F and F' are equivalent if there are nonvanishing functions h j(z) on U j such that fi - I 0 hj 0 fj = R - 1 0 hk 0 fk on Uj n Uk. This is equivalent to

or Uik} = {fjd . b{h;I}, that is, Uik} is cohomologous to {.Ijk}. Thus an equivalence class of bundles defines an element [{fjdJ E HI(au, l!l*(or .@*») s;;; HI(M, l!l*(or .@*». Conversely it is easy to construct a bundle from an element of HI(M, l!l*(or .@*». Thus we have a 1-1 correspondence between equivalence classes of C l bundles and classes in HI(M, (!)*(or .@*». We shall always identify equivalent C l bundles and we call Ht(M, l!l*(or .@*» the group of c i bundles over M. It has a natural group structure if we define F· G = {fjkgjk} where F = Ujk} and G = {gjk}. We construct an important invariant of C I bundles. For any germ of a holomorphic function!, e 2l1i / E l!l*. Thus we get an exact sequence

o--+ 7L --+ l!l --+ l!l* --+ 0, where 7l.. is the sheaf of germs of locally constant integer valued functions on M. We also have the following commuting, exact diagram:

O~r~[~u~O 0---+ 7L - - - + l!l - - - + l!l * - - - + 0,

64

SHEA YES AND COHOMOLOGY

since lJ7 c

~

and lJ7* c

~*.

This yields the exact commutative sequences

... _HI(M, l)-HI(M, lJ7)----;;--+HI(M, lJ7*) lJ'

_H2(M, l)------.H 2(M, lJ7)- ...

···_HI(M,l)-HI(M,~)-HI(M,~*) IJ'

_H2(M.l)_H2(M,~)_···.

Now ~ is a fine sheaf, so Hq(M,

c5* : Hl(M,

~*) -+ H2(M, l).

DEFINITION

5.3.

~)

= 0 for q ~ 1. Thus

c5* is an isomorphism

c(F) = t5*(F) is the (first) Chern class of F.

We remark that c(F) = c(G) if and only if qJ(F) and qJ(G) are equal in Hl(M, .@*) where qJ : Hl(M. lJ7*) -+ Hl(M, .@*) is induced by lJ7* c .@*, where F, G E Hl(M, lJ7*). Hence F and G are difJerentiably equivalent; that is, there are nonvanishing differentiable functions hj such that {ht} = {hjgjt h;;l} where F = {hk}' G = {gik}. Thus: 5.1. The Chern class c(F) of a complex analytic represents the differentiable equivalence class of F.

PROPOSITION

el

Let us give an explicit description of c(F). If F = {ht}, hj . h,.

bundle

.hi =

1

and logJij

+ loghk + IOgJki =

2ni Ciik,

where i = )-=---1. Then c(F) = {Cijk} E H2(M, l). Next consider C" bundles. Let n ~ z and F be a en bundle defined by {fjk}. Let (D be the sheaf over M of germs of matrix-valued holomorphic functionsJ(z) = J;(z) with detJ;(z) "# 0 where the module operation is matrix multiplication. Note that the operation is not commutative. We cannot define the higher cohomology groups of (D but we can define the following objects: Let dlt = {V j} be a locally finite open covering of X. A O-cochain CO = {h}, Jj E nV j , (D) is a set of sections of (D over V j ; a l-cochain c l = {hd where hk E nVjk' (D); a l-cocycle c l is a l-cochain such thathiz) = hj(z) . Jjk(Z) for z E Vi n Vj n V k . Let Zl(dlt, (D) be the set of all l-cocycles. We note that Zl(dlt, (D) is not a group. DEFINITION 5.4. We say that {fik} and {gid E Zl(dlt, (D) are equivalent if there exists {hj} such that gik = hihkh;; I. Let HI(dlt, (D) be the set of equivalence classes of l-cocycles. We define HI(M, (D) = lim HI(dlt, (D). cfI

5.

65

VECTOR BUNDLES

PROPOSITION 5.2. Each element of Hl(M, 6)) represents an equivalence class of complex analytic en bundles.

Proof

Left to the reader.

We now list some methods of forming new bundles from old bundles. Let F be a en bundle, G be a em bundle, q[ = {U i} a trivializing covering and F = {fik}, G = {gik}' We define the following new objects: (1) Whitney Sum Fffi G. the cocycle {hik} where

This is a

hik(z) =

Tensor Product F®G. This is a hik(z) =Jjk(z)®gik(Z). Recall that hjk --

(

hll 21

h21 11

•••

h~~

...

by

(~k g~J.

(2)

hll 11

en + m bundle which is defined

...

enm bundle defined by {hik} where

h

11 nl

•••

hll) nm

,

It::

and here hjtp", =fjkpgjk",. A point in F®G has coordinates (z, e}l, "', qm, .. " ejm), where (z, e1) and (z, ek) are identified for z E Vi n Vk if and only if

ejA =

L fjkP(Z)gjkizKf"'.

(3) Dual bundle F* of F. This is the bundle defined by {Jjt} where fit = (fjk 1)' = Uie)' which is the transposed inverse of Jjk' Then (z, n") is identified with (z, et P) if and only if

ej« = L fjkP(zKt P = Sometimes we write

L ffj ..(zKt p.

et.. for ej«. Then we have n

(t.. = P=1 L Hk. (Z)(:p. (4)

Complex Conjugate F of F. This bundle is defined by the cocycle

{fik}' Let us now define subbundles and quotient bundles. Suppose that, by a suitable choice of q[ = {U i} and of fibre coordinates G, the matrices Jjk in the l-cocycle {fik} defining F can be written as follows:

66

SHEAVES AND COHOMOLOGY

Hencej!tp(z) = 0 for I ~ f3 ~ m, m + I ~ ex ~ n. Thus ej = :Lil=m+tf/t/i cf for ex> m, and if ef = 0 for f3 > m, then ej = 0 for ex > m. Let F' = u UJ X em where em = {(ej, "', ej, 0, . ", On £; and we identify (z, e) and (z, ek) if j = Ajk(zKk' Then F' is a subbundle of F. The quotient bundle F" = FIF' m bundle defined by the l-cocyc1e {C jk }. is a

e

en,

en-

DEFINITION 5.5. A holomorphic (or differentiable) section of F over V£; M is a holomorphic (differentiable) map cp: z -+ cp(z) of V -+ F such that 1lCP(z) = z where F is a holomorphic (or differentiable) en bundle. We see that locally cp is a set of n-functions. Since local sections and germs of sections are defined, we get a sheaf of germs of sections of F. We denote by t9(F) (or !7)(F» for sheaf over M of germs of hoi omorphic (or differentiable) sections of F. Then locally, t9(F) = t9IUj E9'" E9 t91 Uj (sum n-times), where t91 U j means t9 restricted to Vj and t9 z(F) = t9 z E9'" E9 t9 z (n-times). We now review tangent bundles and tensor bundles. Let M be a complex manifold and {U j} an open covering of M with coordinate patches with coordinates (z), .. " zj) on V j • A (holomorphic) tangent vector at z is an element of the form v = :L:=1 ej(olozj). It is easy to see that the set Tz(M) of all complex tangent vectors at z is a complex vector space Tz(M) ~ If z E Uk another chart at z, then we identify :Lil=l Cf(818zC) with :L ej(818zj) if

en.

I jk/i z II

()

8zj

=::;--p. OZk

This is a linear identification so the vector space structure of Tz(M) is well defined. The set T(M) = UzeM Tz(M) is a complex analytic vector bundle defined by the l-cocyc1e {fj~/i(z)}. T(M) is the holomorphic tangent bundle of M. T(M) is the conjugate (holomorphic) tangent bundle of M. And !/(M) = T(M)E9 T(M) is the (complexified) tangent bundle of M. Then e, the sheaf of germs of holomorphic vector fields, is t9(T(M». If M is a differentiable manifold with local coordinates (xj, . ", xj), then the (complex) tangent bundle !/(M) = UxeM!/iM), where

The real tangent bundle !/JI,CM)

= UxeM!/ xlR(M), where

The relation is !/(M) = IT IR(M) ®IR e, where e is the trivial (complex) line bundle over M considered as a real bundle.

5.

67

VECTOR BUNDLES

Let T*(M) be the dual bundle of T(M). If Tz*(M) the transition relations are

= {(ejt> "', ejll)}, then

We use the following notation: An element u E ~*(M) shall be written u = L" ej" dzj, where dzj(a;i}Z~) = b p as an element of T.*(M). Briefly let T = T(M), T* = T*(M). A tensor bundle is a bundle of the form T ® ... ® T® T* ® . " ® T ® ... ® T*.

We denote T®'" ® T = (® T)P for the p-fold tensor product of T. We remark that T is not a hoi om orphic bundle so (® T)P ®(® T*)q is a holomorphic bundle but (® T)P ®( ® T*)q ®( ® T)' ®( ® T*Y is only differentiable. A holomorphic (differentiable) tensor field is a holomorphic (differentiable) section of a tensor bundle. We now give a brief treatment of differential forms. 5.6. A differential form of type (p, q) (or a (p, q)-form) over an open set W~ M is a differentiable section cP: z -+ [z, q> jill ... "pp, ... P4(Z)] of (® T*)P ®(T*)q over W such that the fibre coordinates CPj""""pPI"'P q are skew-symmetric with respect to OCI ••• OCpPI ••• pq • If P = 1, q = 0, cp(z) = 1 cP j..(Z) dzj. In general, we represent the (p, q) form as follows:

DEFINITION

L:=

1 cp(z) = - ,-,

L

p. q. """, IIp 111 ..... Pq

CPj"I'''''PP''''Pq(z) dzj'

1\ ... 1\

dzjP

1\ ... 1\

dz~q,

where dz P, = dz P, and " 1\" is the wedge product of skew-symmetric forms and satisfies for example, dz" 1\ dz P = -dzfl 1\ dz". [Note: We write z" = z" = Zii.] If M is only differentiable we still have !/(M) and !/*(M). Then if W is

open in M, we make the following definition: DEFINITION

5.6'. A differential form of degree p over W 1 cP = p!

L CPj"I"'''p (x) dxjl 1\ .. , 1\ dxjp

is a section over W of !/P which is skew-symmetric in the indices OC1 •• , oc p . If cP is a p-form and x is a q-form, we define the wedge product of forms

68

SHEAVES AND COHOMOLOGY

For example, if cp

= t L CPap dx" CP" 1/1

" dxP and 1/1 = L I/Iy dxY, then

1

= 2" L CP,.pl/ly dx"

A

dx P " dx Y

1 = -3! "L.. X,.py dx"" dx P " dx Y ,

where

Hence

DEFINITION

5.7. If

then the exterior derivative dcp is

1 din . .=, .- , p.

L

acpJ'''' ... ,. P d x~ !\x~

CI.Cl:I,° .. ,CZ:p

1

V

J

J

A

d x":'"···,, dx,,:p N

N

J

J

(1)

L

= ( p+ 1)',,,o,,,,,,.p I/Ij,.o"·"p dx? " '" " dxj",

where

1/1"0"'"'' = (a:"o)CP""""" - (a:,..)CP,.O,.l"'"'' + ...

+(

-l)p(a:"" )cp,.o ... ,.p- •.

5.3. dcp is well defined (that is, the definition is independent of the choice of local coordinates Xj)'

PROPOSITION

Proof We write out the proof for p = 2 and leave the general case to the reader. So consider cp on Uj

(i

= t L ({) jl1P dxj "

dx~ =

t

L ({)U,. dX; "

dx:

Uk' We want to see that

acpj,.p ZL..a Y

1- "

Xj

dx~J " dx~J A dx~J =

acpu,. 2L..a.

.1 "

Xk

dx·k " dx kA " dx"k'

(2)

5.

69

VECTOR BUNDLES

The following rules of transformation are given to us. ox~ ox~

CPj«P =

L o"
l,p

j

The sum of the first two terms is zero, so q=

i

2"

LoCPup ox~ d' ox~ dx' A - - - x', A -

ox~ dx'P

II

OX kP

OX~ J

axIl:J

J

-

OX~

J

J

J

Q.E.D,

and (2) is proved.

For other types of tensor fields such a "nice" operator does not exist. To correct for the difficulty one introduces the idea of a connection. REMARK.

PROPOSITION

Proof·

ddcp = O.

5.4.

ocp« •.. ' «p d xII d cP = -1 ~ L.. p! OX"

A'"

A

dd cP = -1 '" L.. 02cpII' ' 'II p dx P A dx ll p! oxPaxIl

dXIIp.

A ' "

A

dx tlp

=0, since 02CPII,oo'lIp/oxfJox" is symmetric in PROPOSITION

5.5.

Easy.

p and dxP "

dxtl is skew-symmetric.

If cP is a p-form, then d(cp "

Proof

0:,

.p) =

dcp "

.p + (-I)P cP

A

d.p.

SHEAVES AND COHOMOLOGY

70

THEOREM 5.1. (Poincare's Lemma) Suppose that ap-form cP, p ~ 1, satisfies dcp = 0 on a neighborhood U of 0 = (0, ... , 0) s;; RII. Then there is a (p - 1)-form t/I on a neighborhood W, 0 E W S;; U. such that cp = dt/l (locally dcp = 0 if and only if cp = dt/l for some t/I).

Proof

I Let cp = cp(x) = -, Lcp.. ,..... (x) dx'"'

p.

A ••• A

P

dXllp.

We fix p and prove the theorem by induction on the dimension n. The first step is the case n = p, and cp = CP12 ••• p(X) dx l A ••• A dx'. Define t/I(x)

= g(x) dx l

A ... A

dX,-l

by

t

xp

g(XI, .. " x P) = (-I)P-I

CPl'"

,(x l •.. "

X,-I, t) dt,

where W is a star-shaped neighborhood of 0 and (Xl, .. " x P) if x E W so is the line joining 0 and x). Then dt/l

ag(x) = l.J ~ - - dxll. A dx I axil.

1\ . •. 1\

og(x)

= - - dx P 1\ dx l

ox

1\ ••. 1\

dxP

dxP

E

W (that is,

_ I

_ 1

P

= cpo

Now assume the result for n = m - 1, m > p and consider n = m. Then

+

1

cP mil., ... II. p - I dxm

~ l.J

,

(p -1).

1\ ..• 1\

dxII.p-1

ISII.),sm-1

Let

and

./,'I' =

1 )

(p - 1 !

~

i.J

ISII.),sm-1

9(II ••• II

1 p-

dXIl. '

1\ ..• 1\

dxll.p- I •

•

5.

VECTOR BUNDLES

71

Then

dl/l =

1 (p - 1)!

+

ag L -dXIZ

L

m-1

1 SIZ... sm-l

.. = 1

1

L

I

(p - 1).

ax«

CPmlZl "'lZp_1

1\

dX1Z 1 1\

•.. 1\

dX1Z 1 1\

dxm

1\

dX1Z 1

1\ '"

dX"'p-1

•• , 1\

dx lZp - 1•

1 SIZ... S,"-1

Hence

=, L 1

cp(x) - dl/l(x)

m-l

p ...... =1

XIZ1"'lZp(X)

1\

dx" p

=X and

dX = dcp - ddl/l =dcp =0. But 1

dx = p!

L

m-1

aX.. ..... axm I

P

dxm

1\

dx" l

1\ ••• 1\

dx" p

+ terms not involving dxm. So aXIZ I ' 'lZp/iJxn = 0 and this implies that X is independent of x m, X = X(xi,' ", xn- 1 ). By induction X = dq and thus cP = d(I/I + q). Q.E.D. We denote the sheaf over M of germs of p-forms (C"" p-forms) by AP. Then we have: COROLLARY. (to Poincare's Lemma) and only if cP = dl/l, 1/1 E AP-l.

Let cP E AP, P ~ 1. Then dcp = 0, if

Proof We need only remark that d can be defined on the germ level and then use Theorem 5.1. We notice that AO = q), the sheaf of germs of C"" differentiable functions; and cP E A 0, dcp = 0 if and only if cp is locally constant. Thus: THEOREM 5.2.

The following sequence (of sheaves over M) is exact

d A1 O_ _ C _ A o _

where n

d d _ _ "' _ _

= dim M.

THEOREM 5.3.

The sheaf AP is a fine sheaf.

A" _d_ O,

SHEAVES AND COHOMOLOGY

72

Proof Given any locally finite covering {UJ} of M, we have a partition of unity {Pj}, {xlpj(x)~O} s; Uj,

LPj= 1 and

where Pj(x) is a Coo function on M with Pj(x)

~

O. Define a homomorphism

h j : AP-+AP by

Q.E.D. THEOREM 5.4. (de Rham's Theorem) Let dAP be the image of AP under d. Then dr(M, Aq-l) s; r(M, dAq-l) and

HO(M dAq-l) Hq(M, C) ~ dHO(~, Aq 1)· Proof We note that dAp-l cAP is the subsheaf of AP which consists of germs of p-forms qJ such that dqJ = o. A form cp is called d-closed if dqJ = o. A form qJ is exact if qJ = dt/l. Thus Theorem 5.5 says that the closed p-forms modulo the exact p-forms is isomorphic to the cohomology with complex coefficients. For the proof we use Theorems 5.4, 5.3,4.3, and 4.1. Since 0-+ dAp-l is exact for p

~

-+

AP -+ dAP -+ 0

1,

0-+ HO(dAP-l) -+ HO(AP) -+ HO(dAP) -+ H1(dAP-l) -+

•••

is exact. But Hq(AP) = 0 for q ~ 1. Thus,

0-+ HO(dAP-l) -+ HO(AP)

-+

HO(dAP) -+ Hl(dAP-l) -+ 0

(3)

is exact, and Hq-l(dAP) ~ H9(dAP-l) for q ~ 2. Equation (3) gives

H1(dAP-l)

~

HO(dAP)/dHO(AP).

For p = 0 we have d

O-C-A°---+dAo-O. Thus

0-+ HO(C)

HO(AO) -+ HO(dAo) -+ Hl(C) -+ Hl(AO) -+ H1(dAO) -+ ••• -+ Hq-l(dAO) -+ Hq(C) -+ O. -+

Since

Hq(AO) = 0 for q ~ 1, Hq-l(dAo)

~

Hq(C), and Hl(C)

~

HO(dAO)/dHO(Ao)

6.

73

A THEOREM OF DOLBEAULT

for q ~ 2. Thus,

Hq(C)

~

HQ-I(dAO)

~

"" HO(dAq-I) = dHO(Aq 1)

HQ-2(dA I ) ~ ... HI(dAq-2) for q

~

These two statements prove the theorem.

2.

Q.E.D.

REMARK. We have actually proved more. Let f/ be a sheaf over X. By a fine resolution of f/ we mean an exact sequence (al), h

h

h

(al) 0 ---+ f/ ---+ .910 - - + .91 1 ---+ .91 2- - + ...

such that each .9IP is fine. *THEOREM 5.4.

If (al) is a fine resolution of f/, then

HO(X h.9lq- I ) Hq(X, f/) ~ hHO(~,.9Iq 1)

Proof

6.

for q

~

1.

Same method.

A Theorem of Dolbeault (A fine resolution of l!J)

Let M be a complex manifold with covering {U J by coordinate patches where (z~, ... , zj) are local complex coordinates on U j ' Let

1

lp = lp(z) = - ,-,

p.q.

L lp/l., ',,11, "'l1q(z) dz/l., 1\ ••• 1\ dz l1q .

DEFINITION 6.1.

where

and

1

a a

om = -" "L... - 11 "t'IJ,"· m {J q dzfJ "t' p. q. IJ''''I1.{J''''l1 q Z

1\

dzIJ

1\ ... 1\

dzl1q ,

74

SHEA YES AND COHOMOLOGY

where

REMARK.

It is easy to verify that

aa =0,

00 =0.

(1)

From the complex coordinates we get real coordinates x Y defined by

l'

= X2~-1 + i X 21Z • Then

+ i dX 21Z

dz lZ = dx 21Z -

1

dz li = dx 21Z -

1 -

i dX 21Z •

So m

"t'

LEMMA 6.1. d =

Proof

1 = (p +q)!

~ cp 1"

* )'1.

... '''p+q dX Al

"

••• "

dx"P+4.

a+ o.

We easily check that

a + dz -a = dx -a + dy -a.

dz oz

oz

ax

oy

In fact

a)Ii = L211 dX";---A 0 a + 0 = IZ=I L"(dz lZ -aa IZ + dz li = d. Z uZ .. =1 uX

Q.E.D.

:l

COROLLARY.

a a= -0 o.

Proof 0 =dd =(a + 0)(0 + 0)

=a0 + 0 O.

THEOREM 6.1. [Dolbeault's lemma (an analogue of Poincare's lemma)] If a Coo (0, q)-form cp satisfies ocp = 0 on a neighborhood U ~ 1C" of zo, then there is a Coo (0, q - 1)-form 1/1 on W with Zo E W!; U (W open) such that cp = 01/1 on W. First, we turn to: LEMMA 6.2.

Let!(z) be a bounded Coo function on U g(z)

=.=!. fff(O d~ d", (-z

11:

U

where,

= ~ + ;". Then 9 is Coo

on U and og/iJz =f

!;

C.

Suppose

6. Proof Pick

Zo E

75

A THEOREM OF DOLBEA ULT

U and let

..11 = {zllz - zol < s},

..12 = {zllz - zol < 2s} .

Suppose z E ..1 1 • Let II (C) be a Coo function which is identically equal to I in ..11 and is zero outside of ..1 2 • Then

where 11(') = 0 for' ¢ ..1 2, 12(C) = 0 for'

Then we have 9(Z) = 91(Z)

+ 92(Z),

E

..1 1 •

where

de d~ ( ) =.=.! Iffy(Z)r ·

9y z

1t

..

-z

U

For Z E ~1' 92(Z) is hoi om orphic, so we can consider

(oloz)

9iz)

= o. Sinceit = 0 outside of

~1

Let

, - z = W =S + i t

=reiO.

Then 91 (z)

=-

"0

-1 fffl(WW+ z) d s d t = 1t

1 Iff1(re'

- -1t

c

= -

+ z)r dr dO "0 re'

c

~ If fl(re i9 + z)e- i8 dr dO. c

Thus 91(Z) is Coo in z. Then

o 91 = --1 If -;;-=. 0 fl(re'"0 --::

oz

1t

-1 If --:. 0 fl (w

= -

1t

+ z)e-'"0 dr dO

OZ

oz

dt + z) ds W

and

o

-;;-:: fl(W

oz

+ z) =

0 !l-

uW

fl(w

+ z).

76

SHEA YES AND COHOMOLOGY

We use Stokes' theorem. If W is a domain in C with boundary C a Jordan curve, then

t

u dt - v ds

=

e:

fJ w

If cp = udt - vds, then dcp = ( -OU + -OU) ds os ot

+ ~~) ds dt.

1\

dt so

fc cp fw dcp. =

We notice that dw

1\

dw = (ds

+ idt)

(ds - idt) = -2ids

1\

1\

dt.

Hence,

a

1

oz gl(z) = 2ni

If owa fl(w + z) dw

1\

W

dw

- 1 = 2ni

If d [fl(w + z) -;dW] , c

since

= 0 fl dw

1\

ow

dw w

+ afl dw ow

1\

dw w

ofl dw 1\ dw =----ow w Thus, if we set

r

= {zllz - zol = 2e},

y = {zllz - zol = e},

then

= -1.

2m

I fl(w + z) -dw - -2m1. If d (fl -dW) . w 1

W

Taking the limit as e -+ 0 we see that og ogl oz = oz = fl(Z) = f(z).

Q.E.D.

6.

77

A THEOREM OF DOLBEA ULT

Instead of as~uming thatfis bounded we could have proved, using the same proof: LEMMA

6.2. If f(z) is Coo for Izl < R, then

is Coo for Izl < R - e and

O~~) =

J(z)

for

Izi < R - e,

where e > 0 and R - e > O. Proof

(of Theorem 6.1) We may as well assume U = UR

= {zllzl < R, where lal = maxlzlll}.

Then we want to prove that if oqJ = 0,

1

qJ

= q! L qJIII"'II, dzlll

1\ "'1\

is a Coo(O, q)-form on UR , then for any Il > 0, R form 1/1 on UR - t such that

01/1

= qJ on UR-t

dill,

8

> 0, there is a Coo(q - 1)-

(q ~ 1).

Suppose

By this we mean that form qJ does not involve differentials of coordinates for i > m. The proof will be by induction on m with fixed q,n. First we consider m = q. Then Zi

and

=

~ L"

o

-!I-II""I"'q(z) dill {fI

l12:q+ 1 uZ

Hence, (O/Oi'")qJl '"

q

= 0 for ex ~ q + 1. Define

-1 g(Zl, ... , z") = 7r

II

1\

di 1

1\ '"

1\

di q •

78

SHEA VES AND COHOMOLOGY

Theng is Coo on UR-<£/n) and og/OZI = CPI ...q(z). So let t/J(z) = g(z) dz 2 dz". Since O(p/ozrt = 0 for IX ~ q + I, og/OZi = 0 for IX ~ q + 1. Thus, at/J(z) =

t 11=

O~II dz

I

OZ

ll

A

dz 2

A ••. A

A ... A

dz q

= cpo

Now assume the lemma is proved for m

1

~ '" I t... "1'P, "'P q P,p,S,k-l

= -

dz fil

~

k - 1 and consider m = k. So,

A ••. A

dzfiq

and

Thus, 0 = (O/OZIl) CPPI ... pq(z) = 0 for ex ~ k + 1. Define 9pz ... Pq(z) -- --1 ;-

If ~2~2
for 1 ~

P2, ... , P" :$; k -

cP k(lz .•• (lq (Zl ,

...

,

Zk-I " ,

zk+ \ " ... Zn) de d'1 ,

-'I 1. Then gP2"'(lq is Coo on

og and-=O ozrt

UR-£I' for ex ~ k + 1.

Let ,,' =

""

1 I

(q -1).

~

t... 9(lz·"{Jq

dz-(lz A .•• A dz- Pq

P.S,k-l

•

Then

og ow = (q- -- 1 L L dz I)! OZII k-\

p,;S.k-Ia=1

a 1\

dzP'

1\ .•• 1\

dz/l q

6.

79

A THEOREM OF DOLBEAULT

Then (1

= qJ -

ow

=~ L

(111100011.

dzlll

1\

1\

0 ••

dzfJe

P·lliSk-l

is Coo on V R- el . By induction there is a 1/1 which is Coo on V R- el -£2 such that (1 = 01/1. So cp = o(w + 1/1) on U R-el-e2. Q.E.D. We can make an improvement. THEOREM 6.2. (also called Dolbeault's Lemma) If cp is COO on U Rand ocp = 0, then there is a Coo 1/1 on V R such that cp = 01/1 on U R. Proof We first consider the case of a q-form cp with q ~ 2. Let U. = U R - rlv. < r < R. On each Vv+l there is a 1/1. such that ~I/I. = cpo We construct ~1' ~2' ••• by induction on v such that o~. = cp on U .+1 and~. = ~'+I on V •• First set ~I = 1/11 on U 2 = UR-(r/2) Then 0(1/12 - 1/11) = on U2. Hence there is a 0 on UR -(r/2)-£ such that 00 = 1/12 - 1/11. Take p(z) ~ 0, Coo and such that

°

°

0

I. on VI p() z = {

outside V R-(r/2)-2£.

0,

Thus p, 0 are defined on V R' and pO =

{O,

outside U R-(r/2}-Z£ on U I = U R-r.

0,

Define ~ Z = 1/12 - o(p . 0). Then 0~2

= 01/12 on U 3

and

~ 2 = 1/12 - 00 =

1/11

= ~ I on V 1

0

Suppose we have ~l> ••• ,~. such that o~;.=cp on VH1 for A.~v and ~;. = ~HI on U;. for A. ~ v-I. Then 0(1/1,+1 -~.) = Oon V.+ 1 = UR-(r,.+I) and 1/1'+1 -~. = 00 for some 0 on U R-(r/.+I)-£. Define ~'+I = 1/1.+1 - o(pO) where

I, on U. p- { -

0,

outside UR-(r/.+I)-2e.

Then 0~.+1 = cp on U.+ 2 and ~Y+I =~. on U•. Then 1/1 is well defined, Coo on V R, and 01/1 =o~. = cpo Next consider the case q = 1. We assume 1/1. on Vv+l constructed so that 0(1/1.+1 - 1/1.) = on V.+ 1, cp = 01/1. on V.+ 1, and 1/1. is a Coo function where cp is a given (0, I)-form such that ocp = 0. Then 0(1/1.+1 - 1/1.) = implies that

°

°

80

SHEAVES AND COHOMOLOGY

"'.+1 - "'. = I. is a holomorphic function on U.+ I • Let I{II = "'I. Since '" 2 - I{I I = It is hoI om orphic on U 2 let PI be a polynomial such that IpI - Itl < ton UI · Set 1{12 = "'2 - PIon U3· In general, given I{I. such that = ol{l. on U.+ I and 0"'.+1 = cp lJl{I. on U.+ I , we can find a polynomial P. such that Ip. - 1.1 < 1/2v on U. since -I{I. is holomorphic on U.+1. Then II{I.+I -I{I.I < 1/2v on U. so the limit exists,

0"'.

"'.+1

lim I{J. = '" and 00

"'="'.+L(j~-PIl) on

+ h., = 0"'. = cp for all

U.

= "'.

where

oh. = O. So 0'"

v.

Q.E.D.

COROLLARY. Let AO,q denote the sheaf of germs (over M) of Coo (O,q)forms. Then AO = AO,o = PJ and D

D

Ii

Ii

D

0 - - tP-- PJ-- Ao,l-- AO,2 _ _ .. ' - - AO,,,-- 0 is exact.

Proof

COROLLARY.

Proof

AO,q is a fine sheaf. Hq(U R ,tP) = 0 for q

Q.E.D. ~

1.

Use Theorems 6.2 and 6.3 (Ho(U R, oAO,q-l) = oHO(U R, AO,q-I)).

Q.E.D. We can generalize these results. Let cp be a COO(p,q)-form. We have the following sequence of statements whose proofs are similar to the previous proofs [see Gunning and Rossi (1965)]. LEMMA 6.2'. On U R' a (p,q) form cp,q ~ 1 satisfies acp = 0 if and only if cp = 0"', where", is a (p,q - I) form.

COROLLARY'.

Let QP be the sheaf of germs of holomorphic p-forms over M.

6.

81

A THEOREM OF DOLBEA ULT

Then D

C

D

O-QP-AP.o-AP.I- " ' - A P ' " - O is a fine resolution of QP.

REMARK.

Let cp(p,q) be a (p,q)-form. Then ocp(p,q) = 0 if and only if

cp(P,4) = ol/l(p,q

1).

So,

ocp(P.q) = 0 if and only if cp(p,q) = ol/l(P,q

1)

Hence (on U R) ocp(p,q) = 0 if and only if cp(p,q) = ol/l(p -1 ,q), P ~ 1. IfJ(z) is holomorphic on Izl < R, then dg/dz = Jwhere g(z) = J~ g(C) dC. LEMMA 6.3. Let cp be a holomorphic p-form on U R , P ~ 1. Then there exists a holomorphic (p - I)-form 1/1 such that dl/l = cp if and only if dcp = O.

a

Proof d = + 0 so dcp = ocp if cp is hoi omorphic and if dcp = ocp = O. The rest of the proof uses the observation made just before the statement of the lemma. The details are essentially the same as in the previous proofs and are left to the reader. d

THEOREM 6.4. is exact. REMARK.

0- C-

(1)-

d

0 1_

02-

••• -+

0"_ 0

OP is not a fine sheaf.

Now we consider holomorphic vector bundles F over M. Let F be defined by the I-cocycle {jjk(Z)} wherejjk(z) = [fjkP(Z)] ...P=I ....• m. On each coordinate patch (1)(F) = (1) EEl ••• EEl (1) (m times). Let cp be a section of (1)(F) over an open set W!;; M. On W n U j ,

(p(z) = [cpj(z), "', cpj(z)], where cp1 is a holomorphic function on W n UJ' and

L fjk"(Z)CP~(z) for z E W n U n Uk' " By a (p, q)-form cp with coefficients in F over W we mean cp(z) = [cpJ(z), .. " cp1(z) =

j

cpj(z)] where each cpj(z) is a (p, q)-form over W n Uj such that m

cpj(z) =

L fMz)cp~(z) ,,=1

SHEA YES AND COHOMOLOGY

82

as differential forms for Z E W n U j n Uk' We define AM(F) to be the sheaf over M of germs of (p, q)-forms with coefficients in F. At each point x E M, the stalk A~,q(F) = A~,q EB ... EB A~,q (m times). We have: iJ

THEOREM 6.5. 0 -+ @(F) -+ AO(F) ~ AO,l(F) -+ resolution of @(F).

••• -+

oAO,n -+ 0 is a fine

a

Proof We remark on the definition of and leave the proof to the reader. The functions jjk are holomorphic so (818z)jjk(Z) = O. Hence,

So

ais well defined and if cp is a (p, q) form with coefficients in F, ocp is a

(p, q

+ I)-form with coefficients in F. 8cpj(z) =

We note that

L f;k,.(Z) 8cp~(z) + L afJklJ(Z)CP~(z), IJ

p.

and ocp is not well defined on Ao,JI. We also notice that AM(F) = AJI,q®(I)@(F). Remember that cp(JI,O) is a differentiable section of T* ® ... ® T* which is skew-symmetric. We define T* 1\ .,. 1\ T* to be the subbundle of T* ® ... ® T* consisting of those (z, C.. , .....) which are skew-symmetric in <Xl , •• <XJI' Then cp(p,O) is a differentiable section of (1\ T*)P, cp(JI,q) is a differentiable section of (1\ T*)P ® (T*)q, A(JI,q) is the sheaf of germs of differentiable sections of (1\ T*)P ® (1\ T*)q, and A(JI,q){F) is the sheaf of germs of differentiable sections of ( 1\ T*)p ® ( 1\ T*)q ® F.

[3] Geometry of Complex Manifolds I.

Hermitian Metrics; Kahler Structures

Let M be a complex manifold. We want to introduce in M something analogous to a Riemannian metric which is "compatible" with the complex structure of M. Remember that for a Riemannian manifold the element of arc length is given by ds 2 = L g..P dx"dxp• DEFINITION 1.1. An Hermitian metric on a complex manifold M with local coordinates (zj) is given by n

ds 2 =

L

girtl1(z) dzj dz~, ",P= 1

where gjrtp(z) is a Coo section of T* ® '1'* such that (1) (2)

gjrtp(z) = gjPa.(z) P= 1 gjrtp(Z) , ..

L:,

,p

(Hermitian symmetric) ~ 0, and equality if and only if' = 0 (positive

definite). 1.1. Given any complex manifold M, we can introduce an Hermitian metric on M.

THEOREM

Proof. Let I1IJ = {U j} be a locally finite covering of M with coordinate patches U j and let (z} , "', zj) be coordinates on U j • On each Uj we have a metric

the usual Euclidean metric. Let {p j} be a partition of unity subordinate to Cl/i; that is,

Define ds 2 =

L Pj(z)( j

t

),=1

83

dz; dZ;).

84

GEOMETRY OF COMPLEX MANIFOLDS

We claim this is an Hermitian metric. On Uk n

ds 2 =

L gk..lI(z) dZk dzf . ... (J=1

Since OZ~ dz~J = '" _ J dz" '-;:,.. k' .. uZk

Then

± (OZ1) (OZ1)C"C OZk

... /I,A=l

OZ~

±

OZ1 c. 12 > 0 .. ,A=lOZk

11 = 1

if C=/::. O. Thus ds 2 is positive definite and the rest is easy to check. Suppose given an Hermitian metric ds 2 = ). Jl.E{12 ... n I ... jj} by , "'"

( )_(0 gjA/J -

gji/l

L 9 M dzj dz~.

Q.E.D.

Define 9 jAjJ for

9M) 0 .

Then 2'L9M dzj dZ~='LA'/Je(l,2, ... ,jjI9jA"dz1dz~, where, as has been our custom, z~ = zj. We associate to ds 2 a differential form of type (1, 1),w = iL 9j..11 dzj 1\ dz~, where i = ~. REMARK. prefers

Or if one w(C, ,,) = ligM«(j,,~ - "j(~) = -ligj/lii«(~"~ - ,,~C~)

= w«(, ,,). So w is a real form. DEFINITION 1.2. 2 'L 9..11 dz" dz ll is called a Kahler metric if dw = O. Then we call w a Kahler form. M is called a Kahler manifold if we can define a Kihler metric on M. REMARKS (1)

zero.

An Hermitian metric can always be defined, but generally dw is not

1.

HERMITIAN METRICS; KAHLER STRUCTURES

85

(2) Compact Kahler manifolds have many properties similar to (projective) algebraic manifolds. (3) Every algebraic manifold is a Kahler manifold. (Recall that algebraic manifolds are by definition compact submanifolds of IPN for some N.) . (4) There are many nonalgebraic Kahler manifolds. CONJECTURE. A compact complex manifold of complex dimension 2 is Kahler if and only if the first Betti number is ~ven. We know that M Kahler implies that the first Betti number is even (Theorem 5.4, Corollary 2). We have the following facts: (1) If hI (M2) is even, then M2 is a deformation of an algebraic manifold where by M2 we mean a compact complex manifold of complex dimension 2. Thus there is a complex family {Mtlltl < 2} such that M~ is algebraic and Mf = M2 [Kodaira (1964)]. (2) Any small deformation of a compact Kahler manifold Mii is Kahler; that is, if M(j is Kahler, then M7 is Kahler for small enough t. (3) It was conjectured that any deformation of a (compact) Kahler manifold is Kahler. This turned out to be false [H. Hironaka (1962)] for dimension ~ 3. So we make the conjecture for M2. Now we collect some facts about Kahler manifolds. Let U = {zllzl < I}

c

en.

PROPOSITION 1.1. A form qJ = L qJ,,/J dz" only if there is a Coo function! such that

A

dz/J on U satisfies dqJ = 0 if and

02j ( OJ) = "'.LfJ - dz" A dzfJ . oz." ozfJ '

qJ = ooj= 0 L-dz/J ozfJ that is, if and only if qJ,,/J

= (fP!lozQ ozfJ).

Proof (I) If qJ = aGf, then d(oo!) = (0 + o)(oof) =ooo! = -ooo! =0. (2) Suppose dqJ = oqJ + oqJ = O. Since oqJ is of type (2, 1) and oqJ is of type (1, 2), oqJ = 0 and oqJ = O. Dolbeault's lemma implies that there is ",(1.0) such that 0",(1.0) = qJ on U, tIP· 0) = dz". We have

L "'"

0= oqJ = 00",(1·0) = _00",<1,0)

86

GEOMETRY OF COMPLEX MANIFOLDS

and

So

o (01/1 11

01/1,,)

oz" - OZII

OZA

=0

and 01/1(1·0) is a holomorphic 2-form. According to Chapter 2, Lemma 6.3, there is a holomorphic I-form '1 on V such that 01/1(1.0) = d'1 = 0'1. Thus 0[1/1(1·0) - '1] = 0 and there is a Coo function f such that 1/1(1.0) - '1 = of (see the remark after Chapter 2, Theorem 6.3'). Hence,

cp = 01/1(1.0) = 0('1

+ oj) = oof =

oo( -J).

Q.E.D.

Let 2 L gMi dzj dz~ be an Hermitian metric and w = i L gMI dzj " dz~ be the associated form on Vj where the manifold M = uU j . Assume U j = {zjllzjl < I}. THEOREM 1.2. w is a Kahler form (that is, dw = 0) if and only if there is a differentiable function K] on each U] such that w = 00 K j on U j .

Proof

Use Proposition 1.1.

PROPOSITION 1.2.

P" is a Kahler manifold.

Proof pn = Uj=o(U), where Uj = {' I' = ('0' "', en), 'j =F OJ. On Uj then we have (affine) coordinates z·J = (z~J' ... , Z~-1 z~+ I , ... ' J z~) ' where J 'j zj =

',,/'j'

We set K j = log(l

= log

+

L,IZjI2)

"'*J

Cto Ie..I

2) - logle j l2

on U j • Then

K] - Kk = loglek"jI2 = loglz~12

= log z~ + log z~ on Uj n Uk' Therefore ooK] = ooKk on Vj n V k , and we can define a global form w = i oOK j on each V j • Clearly dw = O. If w = ('[. gJcxi/ dzj " dz1, we must show that gj(1.P is positive definite; Hermitian symmetric will be left to the

HERMITIAN METRICS; KAHLER STRUCTURES

l.

reader. Take j

87

= 0 and let z,. = zo. Then Ko =

10g( 1 + ,.tIZIl12),

so

and

Then

where n

(C z) =

L ,IlZ". Il= I

The Schwarz inequality implies that 1('. zW :=:; is a positive definite form. Q.E.D. THEOREM 1.3.

«(. 02(Z. z)1 2hence Equation (I)

Any submanifold of a Kahler manifold is a Kahler manifold.

Proof Suppose that N eM and w is a Kahler form on M. We claim that the restricted form wiN, which we shall shortly describe. is a Kahler form on N. To define the restriction we proceed as follows: On each coordinate patch Vj with Vj n N =1= cp we choose local coordinates (zJ •...• z} • ...• zj') = z~ IV~ ••• w~) with,. = m - n m = dim M n = dim N so that (z~J ' ... ' J ' J' 'J ' ,

Nn Uj={{zJ,···.zj.O,···,O}}.

Suppose 1 cpo ••• cp = -p! '\' L- JIl,

f1.p

(z.J' w.) dz J

ll '

1\ ••. 1\

dz":p J

then 1 '\' cp.Jf1., ••• cp I N = - ,Lp.

IIp

(z.J' 0) dz":' J

1\ .•• 1\

dz":p J

(3)

88

GEOMETRY OF COMPLEX MANIFOLDS

is a form on N since

zj = f}k(Zk' wk),

o=

g;k(Zk, 0),

and on N,

Similar remarks apply to forms of type (p,q). We see that eplN is a form of type (p,q) and that

d(ep I N) = dep I N. Thus we see easily that wiN is positive definite and d(w IN) = O. COROLLARY. Proof.

Q.E.D.

Any algebraic manifold M is a Kahler manifold. M ~ P" for some n.

THEOREM 1.4. If M is compact and Kahler, then the Betti numbers b2k(M);;:: I for k = 1,"', n, n = dime M. Before giving the proof we shall embark on a brief discussion of integrals of forms, specifically m-forms, on a compact differentiable manifold M of dimension m. Suppose Xj = (x], "', xj) is a local coordinate on U j C M. Then an m-form ep can be written

ep = epjI2 ... m(x)dx}

1\ ••. 1\

dxj on U j

= epk, ... m(x) dxl/\ ... /\ dx~ on Uk' and on Uj n Uk.

axil.) dx l dx~J /\ ... /\ dX'~ = det ( -1 J II k

a

/\ ••• /\

Xk

To be able to define

f

dxmk'

(4)

ep we make the assumption that M is oriented. This

M

means that there is a covering of M with local coordinate patches (U j ,x) such that det (axj/ax~) > 0 on Uj n Uk' Any local coordinate (Xv, ev) is positively oriented if det(oe~/i.lx~) > 0 for allj such that Vj n Xv i: . Let {Pj} be a partition of unity subordinate to {U j}.

J ep

f Pj(x)epj, ...m(x) dx) ... dxj. Using Equation (4) and the orientability it is easy to check that f cp is well defined indepenDEFINITION

1.3.

M

= Lj

Uj

M

1.

89

HERMITIAN METRICS; KAHLER STRUCTURES

dent of the choice of covering by local (positively oriented) coordinate patches and partition of unity subordinate to it. We leave this to the reader.

PROPOSITION

If cp = dt/l, then

1.3.

f

cp = O.

M

Proof

t/I =

Ij Pj(x) t/I /x), so

where

Uj = {xjllxjl < rj} and Pj(x) = 0 for x ¢ Wj We will show that each

f

£;

Uj .

d(pjt/lj) = O. To simplify notation let us drop the

Uj

subscripts. Then m

I

Pj t/I/x) =

11=

hix) dx 1

/\ •.•

dx~- t /\ dx~+ 1

/\ ••• /\

dxm

1

and hix) = 0 for x ¢ Wj . We calculate

fu. d(p t/I): j

J

f

f =f

d(pjt/l) =

Uj

Uj

d(Ih ll dx 1 a.

/\ ••• /\

dXa.-l /\ dXa.+l /\ ... /\ dx m )

I (-l)~+ 8h~ dx ' ... dx m 1

Uj

8x~

a.

= I(-l)'z+1 "

f

dX'''' dX~-l dX~+I ... dxm

Ix'i
f

r

-r

ah

-;dx"

or

=0, since

f

r

-r

8h" -8" dx" X

= ha(r) -

ha( -r)

= O.

Q.E.D.

A complex manifold M is naturally oriented. For if (zJ ' ... , zj) are local coordinates and zj = X;~-l + i xr, then the coordinates (x~, .. " X]n) give a covering of M and the determinant of a change of charts is detI8zj/8zfj2 > 0 hence M is oriented and (x] , .. " x;n) is a positively oriented chart.

Proof

(of Theorem 1.4) Let w = i I gall dz~ /\ dz ll . Then

a a n P = (i)n '\' L. g ",11, g a.2/h··· g ".11. dz , /\ dz , /\ ••• /\ dz • /\ dzfJ •

90

GEOMETRY OF COMPLEX MANIFOLDS

We claim

I

w" > O. For

M

(1)"

= (i)"

2: sgn( IXI1 ::: IX"11) sgn(pI I ::: pll )g~tlll ...

a.p

II

g" .. P.. dz ' /\ dz T

•.. /\

dz" /\ dz ii ,

and I ... ...

L sg n ( a. a,. ".a"

I

11) IX

g.IPI ... g.IP ..

ga,p, ... ga .. p" -. .

g

II

liP,

.. ... g

liP..

1 ...

11)

= sgn ( p1 ... P II

1 ...

=

11)

sgn ( Pl'" PII

g,

where g =det(g"p). Therefore,

w" =(i)"n!g dz 1

/\

••• /\

dz ii •

Since

dz 1

/\

dz T

= (dx 1 + i dx 2 )

/\

(dx 1

-2 i dx t /\ dx 2 , = 2" n! g dx t /\ •••

-

i dx 2 )

=

(1)"

/\

dX 211 •

We have assumed (gaP) is positive definite so g > O. Thus SM (1)" > O. If wn = dljJ then JM (1)" = 0 by Proposition 1.3. Thus (1)" =P dljJ. In fact. we claim w /\ ... 1\ (1) = ol =P dljJ for any 1jJ. Since d(1) = 0, d(ol) = 0 and if (1)k = dljJ then w n = (1)k /\ (1)"-k = dljJ /\ W,,-k = d(1jJ /\ w n - k). Now recall that

bu

= dimeH2k(M,

HO(M dA 2k -

C)

1)

= dime dHO(~, A2k 1)'

The facts just proved show that w k E HO(M, dA 2k - I),

w k rf: dHo(M. A Zk-I). Thus, b 2k

~

I.

THEOREM 1.5. If M is compact Kahler and if N c M is a compact complex submanifold. then N is not homologous to zero in M.

Prool

(sketch)

We first prove:

1.

HERMITIAN METRICS; KAHLER STRUCTURES

91

PROPOSITION 1.4. (Stokes' theorem) If W" + I is a compact differentiable manifold with boundary oW n +1 = Mil, then J","+I dl/l = SM" 1/1 for any n-form 1/1 on W= W n + l • Proof We cover W with a locally finite family of coordinate patches {Vj} such that Vj = {xjllxjl < r j } if Vj ~ int(W) and if Vj n M =F cp, then Vj={x)lxjl I, -rj<xJ~rj}' and MnVj={xjlx]=rj}. We

choose a partition of unity {p.;} with the following properties: supp PJ = {xl Pj(x) > O} e Vj if Vj n M = cpo supp Pj ~ {xjl-rj < xj ~ r j , Ixjl < r i , oc ~2}. (3) Pj is Coo, Pj ~ 0 and L Pi = I.

(I) (2)

Then Iwdl/l = LjIwd(Pil/l) = LjIujd(pjl/l). From Proposition 1.3 Iujd(Pk 1/1) = 0 if V k n M = cpo Suppose Vj n M =F cp and 1/1 = L::: I/Ij dxj, /\ ... dxj-l /\ dX~+l /\ ... /\ dxj+1 on V j . Then

and

Foroc=I,

Hence,

f

Uj

d(p J'I' .. 1,)

=

f Uj

p.J'I'J ·"~(r.J' x~J' ... • x'!+ 1) dx~J ... dx'!+ I J J

Thus,

where M=oW. For the proof of the theorem, suppose M n is compact Kahler and N m is a complex submanifold N m eM". Suppose N =oW eM for Wan embedded submanifold. Then if w is the Kahler form on M, 0 < IN w m = Iw dw m = O.

92

GEOMETRY OF COMPLEX MANIFOLDS

This contradiction proves the theorem in this case. Generally if N is homologous to zero, we do not have such a convenient situation. One must change the proof, and we supply no details here.

2.

Norms and Dual Forms

Let OP be the sheaf over M (a compact complex manifold) of hoI om orphic p-forms. Let AM be the sheaf of Coo (p, q)-forms on M. We want to introduce an Hermitian scalar product (cp, 1/1) for cp, 1/1 E r(AM), which makes r(AM) into an (incomplete) inner product space. We introduce an Hermitian metric 2 L gj,.p dzj dz~ = 2 L gap dz" dz P on M. Associated to this metric we have the form w = i L gap dz" 1\ dz P and wn = 2n n! 9 dx 1 1\ ••• 1\ dx 2" as before where X 2a - 1 + i x2a = z,. and 9 = det(gaP)' We denote the inverse (g,.p)-l of (g,.p) by (gi P) = (g,.p)-l, that is,

L giPgp'Y = <>;

and L gapgPY = <>~.

P

P

The length 1'1 of a tangent vector C is given by inner product (c,,,) = La,pg,.pC""p. Let cp(z)

= _1_ L cp p! q!

p dz'"

1(21 = L,.,p gap cafi and

A ••• 1\

dz Pq

1\ ... A

dz Pq

the

q

al'"

and 1/I(z) = _1_ L 1/1 p dZ"1 p! q! al'" q Then at each point

ZE

M we define

= _1_ " glIal (cp , .I.)(Z) 'I' I I i..J p.q. a.II .... JJ

DEFINITION

2.1.

••• gPIJJI •••

gPqJJqcpal "',.q ~ '1'''1 ~ ... JJq'

(1)

The inner product of two forms cp, 1/1 is

(cp, 1/1) =

f (cp, 1/I)(z) w:n. f (cp, 1/I)(z)2"g dx =

M

M

(,) satisfies the following properties: (1) (cp, 1/1) = (1/1, cp), (2) (a1/l + b X, cp) = a(1/I, cp) + b(X, cp), (3) (cp, cp) ~ 0, (4) (cp, cp) = 0 if and only if cp = O. We define

•

IIcpli =J(CP, cp) as usual.

1 •••

dx2n.

(2)

2.

93

NORMS AND DUAL FORMS

THEOREM 2.1. There is a linear map *: r(AM) -+ r(A n - p,n- q) such that: (I)

ron (cp, 1jJ)(z) - = cp(z) " *i/i(z), n!

(2)

*1jJ = *i/i (that is, * is a real operator), **IjJ(p,q) = (-I)P+q ljJ(p,q).

(3)

Proof Before giving the proof let us fix some notation. Let n = dim M. We denote as follows:

and (0(1' .•. , a p ' O(p+1' ••• , O(n) is a permutation of (I, ... , n). For example, = 5 and A2 = (2, 4), A n - 2 = (I, 3, 5). Similarly for Bq = (Ph···' Pq), Bn- q = (P q +1> ••• , Pn). Then with this notation we write a (p, q)-form

n

(3)

where dzAp =dza,

" ... " dZ«p, and so on. We denote .1,ApS, 'I'

= "giilAI ••• gil.P •• I, L'I'AI",A

__ p !ll"'!l.'

A,!l

Then

L L (-IYQIjJ Apli, dz B , " dz/l p = L L (i/ih,::t, dz B, " dz::t,.

=

Thus, (i/i)S,A p = ( -1)PQIjJ ApBq·

We can now write Equation (1) as

(cp, 1jJ)(z) =

L

Ap,B,

CPA

II

P'

IjJApB, = (-lyQ

L

Ap,B,

CPA B i/iS,A P • p,

Remember

Then we define *,1, = (i)n( _1)+
(4)

94

GEOMETRY OF COMPLEX MANIFOLDS

First we prove Theorem 2.1 (I); that is,

_

cP "

*'"

wn = (cp, ",)(z) n!.

If", is as in Equation (3), *:r. 0/

= (i)"( _

1)tn(n- I I +qn ,L. ,L. g ApAn - pB.Bn - • 0/ :r.B.Ap dz An - p

Ap

"

dz lin - •.

(5)

B.

Let

where Ap

= (A.I , "', A.p ), Nq = (VI' ••• , vq) with the usual conventions. Then

m" *:r. =

(+ iii), g ApAn-pBqBn-q'l' :r.B.Ap(fl dz"p ~ 't'l\ p 'N q

.."

Y'

1\

dz lil • " dz An - p " dz B .. - q

=(±in)(-l)q(n-pll:("')dz"p" dz Au - p " dz lilq " dz Bn -., where (±in) = i"( _I)t(n-I)n+qn from (4). Now, dz"p " dz A.. - p =F 0 if and only if Ap = Ap; similarly we need only consider Nq = Bq. Thus m"

T

*x = in(_l)tn(n-II-qp, m XB.A pg 0/ L. TApB.o/ ApA .. _pBqBn_q dzApAn-p" dzBqBn-q.

Recall as in the proof of Theorem 1.4

f3. g,

gAnAn-pBqBn-q = sgn ex' sgn dzApAn-p = sgn ex dz l dZ BqBn - q = sgn

f3 dz I

.. , "

dz n,

" ... "

dz",

"

where sgn ex = sgn ( 1 ... n ) and so on, and ex l •.. (Xn dz l

" ••. 1\

tizn " dz T

" ..• "

dz ii = (_l)n(n-I)/2 dz l

"

dz T

" .•. "

dz n

Thus, cP " *iJi

=

in( -1)pq" L.

CPA II Bq iJiBqA pg dz· " dz T

= 2n(cp, "')(z)g

dx l

"

'"

"

1\ ..• 1\

dz ii

dx 2 "[by Equation (4)]

wn n!

= (cp, ",)(z)· - [by Section 1, Equation (5)]. Thus Theorem 2.1(1) is proved. For Theorem 2.1(2), *,1, = (_ i)n( _1)n(II- I )/2 + pll , g Y'

L

_

,1,BpAq dzilu -. " dzBn- p

BpBI1-pAqAn-q'"

"

dz ii •

2.

95

NORMS AND DUAL FORMS

since 111(n - 1)

+ pn + 11 + (11

-

q)(n - p)

+ pq =

+ /1 2 + 11 - nq 1) + /1q (mod 2).

111(n - I)

== 1n(11 -

Last we must check Theorem 2.1 (3). Before doing this we make an assumption which we could have made for 2.1 (1) and 2.1 (2), which simplifies the calculations. We must check 2.1(3) pointwise and at any point zo. We may assume by a change of coordinates that gaP(ZO) = bap . This will not be true in a neighborhood of zo; we only assume it at zoo But since we only check 2.1(3) pointwise, each time we verify it at a point we may assume that gaP = ()aP' Then

=

1/1 B

and I/IBpAq =

p

sgn(p),

A• . Thus

=(i)n(_1)fn(II-1l+IIPsgn(IX)", _ ( *"') 'I' An-.Bn-p 'l'BpA.

p

at zo and

*(*,1,) _ = (i)n(_l)tn(n-l)+n(n- q) sgn(B"-pBp)(*,,,) 'I' BpA. A A 'I' An-qBn-p n-q q

= (-l)p+q,I,'I' B.A. -.

Q.E.D.

* has one more property: PROPOSITION

Proof

But also, cp

2.1.

iii

1\

* 1/1 = 1/1

1\

* iii for

cp, 1/1

E

r(AM).

* I/I(z) = (cp, I/I)(z)w"/n! = (1/1, cp)(Z)W n/l1! = 1/1 1\ * iP(z). (Remember that w" is real.)

cp

1\

1\

* I[i = iii

1\

* 1/1.

Q.E.D.

We next define the adjoints of 0, 0, and d.

DEFINITION

2.2.

91/1 = - *o( *1/1), 91/1

= -

*(0 * 1/1), and

~I/I =

-

*d( *1/1).

96

GEOMETRY OF COMPLEX MANIFOLDS

It is easy to see that

8: r(A(P,9» _ _ r(A(p,9-l),

PROPOSITION

2.2.

9:

f(A(p,9»-r(A(p-I,9»,

~:

r(AP)-f(AP-l).

Assume M is compact. Then

(OqJ, 1/1)

=

(qJ, 81/1) for qJ E f(A(P,9- 1 », 1/1 E r(A(p,q»,

(oqJ, 1/1) = (qJ, 91/1),qJ E f(A(p,9», 1/1 E r(A(P+I,9», ~I/I),qJ E f(Ar),

(dqJ, 1/1) = (qJ,

1/1 E r(A r+I).

(Integration by parts; we only prove the first equation.) X. = cp " * i!i is of type (n, n - 1). Hence, oX = 0 and dx. = ox.. Thus, Proof

o=

f

J

M

ox. =

f

M

ox.

=

f o(qJ " *i!i) M

So

(OqJ, 1/1)

= - fM qJ" ** o(*i!i) = -

f qJ" *(*0(*1/1» M

=

f CP" *(81/1)

= (qJ, 81/1).

Q.E.D.

M

If cP =

L CPp dz P, then ocP = 21 ~L. (OqJlI OZi -

OCP«)

i

11

ozP dz "dz.

Thus the coefficient (ocp);p is similar to a .. rotation." We claim that 8 acts like a divergence.

2. PROPOSITION

2.3.

For

t/I E r(A(p,q+l»,

(8t/1)Ap p, ... P. = ( -l)P+ 1

t

(~+ 0 log g)t/lJ.ppp, ... Po.

P= 1 oz{l

REMARK.

L ",P dxP• Proof

97

NORMS AND DUAL FORMS

In Euclidean space div

iJz{l

t/I = L (o",p/ozP)

for a I-form, '"

=

(of the proposition) By definition (oq>, "') = (rp, 8",).

For convenience we omit the index Ap and use the notation

Then

Since

where

Pi means " omit" Pi' Then . (orp, "') =

~ (-l)P fLOp rp't""q ",PII, "'lI qg 2" dx 1 ••• dx 2n q. M

= -( -l)P ~ q.

J L rp"

",11. 011("'PP, ... P4g)2" dx 1

•••

dX 211

M

(integrating by parts). Thus,

Q.E.D. Next we make: DEFINITION 2.3. 0 = 08 + 80. 0 maps r(A(p,q» into r(A(p,q» and is called the (comple9 Laplacian. 0 is a partial differential opera..tor and we want to r (C:"r'f\e;K , 'L c \ . ( compute its principal part.

II

~*fI.

'1_1

H . ,VI

.

l

t.

r

0-,

GEOMETRY OF COMPLEX MANIFOLDS

98

PROPOSITION 2.4. '/') - (0 If' AB -

Let A

= (ex l , " ' , ex p ), P= (PI' ... , Pq ). Then

8

gO" ~ 8Z A 8 Z 0 A.v~l "n

~

2

+ "~

+

(8 h 8 MN )'

A.M,N

AB

+ hMfiJX AB

CPMfiJ A Z

8) CPMN 1;: 11.2

L k'1: CPMfiJ'

M.N

Thus the principal part of D is

82 8Z A 8"' A. ~ v= 19 2 n

"

REMARK.

"A

If gVA = boA, then

1 =--L -a ) 4k~1 ox 2n

(

k

Proof

(of the proposition)

2 •

From Proposition 2.3,

(9cp)AIPl .. Pq-, = -( -1)P Lap cpApPfJ, ... Pq-'

+ order zero terms.

We note that fJ (J.." m~'" pp, ... pq- ,=

"g~').' ~

... giiP . gii'P, ... ap.." m .. , , ·~~t

+ terms of order zero. Thus" lowering indices," (9cp)A pp, .. Pq_' =(_l)P+l LgiiPOpCPApJ;P, ... P._, p,l'

+ order zero terms. Then

+ terms of order ~ On the other hand,

I,

.. ·

2.

99

NORMS AND DUAL FORMS

Thus, +1" -P (90CP)A L.gll Op( 0CP)ApilP, ... P. p p, .. p.=(-l)p

+ terms

of order:::; 1

L gilP{op Oil CPApildh ... -

= -

(7)

op 0PI CPA pllP2 ..

+ ... }.

P.1l

Many terms cancel when Equations (6) and (7) are added yielding

(DCP)A pl/lil2 ... iI. = -

Lg

P.il

op Oil cP Apl/, .•. l/.

ft/J

+ lower order terms. Li

Similarly we define PROPOSITION 2.5.

=

Q.E.D.

oa + 80, 6. = db +f>d and prove:

L giiP 02/0ZPozii + lower order terms /). = - L 2g ilP 02jozP ozii + lower order terms.

LJ = -

(.6 is the real Laplacian.) Proof

Left to the reader.

The operators D, D, and .6 are second-order partial differential operators. Since d = 0 + 0, b = 9 + we get

a,

.6=~+f>d=~+~9+~+~+~~+~ = D

LEMMA 2.1.

09

+ 0 + 09 + 90 + 09 + 90.

+ 90 is a

first-order operator;

+ order zero terms. (09cp)a.o ... a.pl/, ...

= (-1)P+l

+

L gilP{oa.o OpCPa.,a.2·'· -oa., OpCPa.Oa.2'" + ... }

Il,P

(8)

lower order terms.

(9)

(09cp)a.o ... a.pPI ... = ( -l)p L gilP op{oa.o CPa., ... ;;i, ... - oa.,CPa.Oa.2 ... IlP, ... + ...} ilP

+ lower order terms. When we sum Equations (8) and (9) the second-order terms cancel.

Q.E.D.

100

GEOMETRY OF COMPLEX MANIFOLDS /:::. = D

COROLLARY.

+ 0 + first-order terms.

For our purposes the following theorem emphasizes the most important fact about D, D, and /:::.. For the definition and standard facts about elliptic operators we refer the reader to Palais (1965) or Hormander (1963). THEOREM 2.2.

Proof

3.

D, D, and b. are strongly elliptic partial differential operators.

Q.E.D.

g"P is positive definite.

Norms for Holomorphic Vector Bundles

The main purpose of this section is to extend the results of Section 2 to vector bundles. Let F be a holomorphic vector bundle over the complex manifold M and r(A(P,q)(F» the space of COO (p, q)-forms with coefficients in F. Let {fJkv} be a l-cocycle defining F on the coordinate covering "It = {U j} of M. Then locally cp E r(A(p,q)(F» is given by cp = (cpJ(z), "', cpj(z» on Uj ,

where CPJ(z)

1 p.q.

= -,-, L Cpj'" ... p,,,. dz'"

II ... II

dz P,

II " ' ,

m

CPJ(z) = L fj~y(z)cp~(z). y=

1

By definition acp = (ocpJ(z» which is well defined since O/Jh(Z) = O. Let 2 L gAil dzJ. dzil be a given Hermitian metric on M. An Hermitian form on the fibres of F is defined by specifying on each UJ a positive definite form

L ajAil(zKJr;

aj(e, C) =

A,Y

such that ajAv(z) is Coo and aie,

C) = a" (", ,,) where" = /"i . C.

REMARK. Such forms always exist. Let {Pj(z)} be a partition of unity subordinate to the locally finite covering "It and set aCe, O(z) = LjP/z) LA= 1 IeJI2. Let cp, 1/1 E r(A(P,q)(F», cp(z)

= (cpJ(z», I/I(z) = (I/IJ(z».

Then we define a Coo function (cp, I/I)(z) by m

(cp, I/I)(z) =

L A,I'= 1

ajA;;.(cpJ, t/I'J)(z),

3.

NORMS FOR HOLOMORPHIC VECTOR BUNDLES

101

where (cp;, I/I~)(z) is the product of (p, q)-forms at z. (See Section 2.) If M is compact we define

I (cp, I/I)(z) "Ironn.

(cp, 1/1) =

(dime M

M

= n)

We want to define the adjoint 911 of 0 (with respect to the metric a). We want to solve (ocp, 1/1) = (cp, 9/J 1/1)

for 9 11 , Let cP

E

r(A(p,q)(F)), 1/1 E r(A(p,q+l)(F»). Then m

t

=

L

_

ajAiilfJj"

A,,,= 1

*1/17

defines a differential form of type (n, n - 1) and hence dt is a 2n-form. Then

But

Thus,

Let ( ajiiA) = ( a jiiA )-1 ;

. t hat IS,

m

1:"

"iiA L. a j aj).• -_ u •. A=l

Then (*9 a l/l)" = -( -ly+q (9 a l/l)" = -

Hence:

~ a~"(a ~ ajvl(.I/I})),

~ a~".(a ~ ajvl(.I/Ij»).

GEOMETRY OF COMPLEX MANIFOLDS

102

If we expand this out we get (,9 a t/!)j

= -(*o*t/!)j - L a;~*(oajVA II *t/!j) A,V

= (8t/!)j

+ terms of order zero.

The reader can then easily verify the following: Let

D. = 08. + 8.0, then

D. = 0 + terms of order

~

I,

Hence: THEOREM 3.1. D. = - L gfJa o2jiJzaoz P + lower order terms and hence, D. is a strongly elliptic second-order operator.

4.

Applications of Results on Elliptic Operators

For the results about elliptic operators on manifolds that we need Palais (1965) is a good source. First we fix some notation: Let fiJq = qA(p, q)(F» where F is some holomorphic vector bundle. We drop the subscript a and let 8 = 8., 0 = Oa. PROPOSITION 4.1.

0 is self-adjoint, that is,

(Ocp, t/!) Proof

=

(cp,Ot/!).

(0 cp, t/!) = «0,9 + 80)cp, t/!)

+ (ocp, ot/!) (cp, (08 + 80)t/!)

= (8cp, 8t/!) =

= (cp, 0 t/!).

Q.E.D.

The following is the fundamental result about elliptic operators on compact manifolds: Let

THEOREM. (a) dim J{'q < + 00. orthogonal direct sum; so every cp Cp=1'/+(.

(b) E fiJq

fiJq = J{'q EB OfiJ P, where EB means has a unique representation,

(=Ot/!.

4.

APPLICATIONS OF RESULTS ON ELLIPTIC OPERATORS

PROPOSITION 4.2. Proof

Dq> = 0 if and only if oq> = 8q>

= O.

(Oq>, q» = (oq>, oq» + (9q>, 9q» = lIoq>1I 2 + 119q>f.

(t!, oq»

103

Q.E.D.

= (91'/, q» = 0,

= (0". 1/1) = 0, (oq>, 91/1) = (ooq>, 1/1) = o. (", 91/1)

Thus, yt'q, o.!l"- 1, and 9.!l'q+ 1 are orthogonal and .!l'q = Jf(q Ef) D.!l'q = Jf(q Ef) o8.!l'q EEl 80.!l'q £ Jf(q £ .!l'q.

ffi o.!l'q ffi :}.!l'q Q.E.D.

We next have the important theorem relating cohomology groups and harmonic forms. THEOREM 4.1. [Kodaira (1953)] Let F be a holomorphic vector bundle on a compact complex manifold. If QP(F) is the sheaf of germs of holomorphic p-forms with values in F, then

Proof

Recall HP(M, QP(F)

~

f(oA (P.,-I)(F))/of(A(P.,-I)(F)) and

f(oA(P.q-l(F)) = {q> I q>

Let Z;; (.!l'q) = {q> I q>

E

E

f(A(P·q)(F)), oq> = O}.

.!l'q, oq> = OJ. Then

We claim (1)

The inclusion

104

GEOMETRY OF COMPLEX MANIFOLDS

is obvious. Let cP E ZlJ(!£q), cP

= '1 + 01/1 + 90'. Then ocp = 0 so 090'

=0

(090', 0') = 0 (90', 0') = 0 and 90' = O. Thus Equation (1) is true and implies the theorem. COROLLARY.

dim Hq(M, OP(F» <

Q.E.D.

+ 00.

EXAMPLE. Let M"=C"/G be a complex torus. If Z=(Zl,···,Z") are 1 dz'" A dz« defines a 2-form on M" associacoordinates on C", then w = i ted to the metric on M" lifted from the Euclidean metric ofC". If z'" = X 2",-1 + i x 2'" then 0 = -! If::'l (O/OXk)2. Let

L:=

1

""-p!q!L,"""'I,,·,.P/ll···/lq dZ"'1 (f)

-

- - "

(f)

A ••• A

dz/l q.

Then Dcp = 0 if and only if Lf~l (O/OXk)2 CPA.B necessarily constant so

= O. Such solutions CPA.B are

THEOREM 4.2. [Serre duality; see Serre (1955)] 4.1. Let F* be the dual bundle of F. Then

Let F, M be as in Theorem

Hq(M",OP(F»

~

H,,-q(M",O"-P(F».

Proof Let w = i L g,./l dz" A dzIJ be the form associated to a metric on M. If F is represented by {fjkJ on o/i = {U j} (that is, (j = L~= 1 fjkv (~ on U j n Uk) then F* is represented by {fZjl} (that is, '1jl = I~=l f'k j '1u on U j n Uk) and I"njl 'fik" = br. If aiC 0 = (jaj~j (matrix notation; At = transpose of A) is an Hermitian form on F it is easy to see that a transforms as follows: ak = f jkaj' fjk' t

-

(2)

We define a conjugate linear map p: F ~ F* by p«(j) = I ajl.G, that is, E F* acts on '1j by p(U('1j) = I ap..• '11(~. From Equation (2) we see

p«(j)

4.

APPLICATIONS OF RESULTS ON ELLIPTIC OPERATORS

lOS

this is well defined. Using p we introduce a metric on the fibres of F* by

b(", ,,) = a(p -1", p -1,,). It is easy to check that the matrix of b(", ,,) is

= ail =

bj

(a~.t) on V j

•

Let

.1fp,q(F) = {lP IlP E r(Ap,q(F», DlP JIf"-p,,,-q(F*)

= O}

= {lP IlP E r(An-p,n-q(F*», DlP =

If F were trivial we could define a map,

r(Ap,q) - --+ r(Aq,p) - -*+ r(An-p,n- q)

lP - - + iii - - + * iii = #-lP· In general we define #- lP by

(#-lP)jJ. = LEMMA

4.1.

L ajlv(*lPj) = P *. 0

#-: r(AM(F) -+ r(An-p,n-q(F*» satisfies #-

0

#-

= (-l)p+qid,

where id = identity.

Q.E.D. Recall

and since aj is nonsingular

(8lP)j=O

ifandonlyif

o(~ajvji*lPj)=O.

Conjugate this to get

that is

Thus

Jlfp,q(F) = {lP I OlPJ = 0, o( #-(P)jl = A},

O}.

106

GEOMETRY OF COMPLEX MANIFOLDS

and ytn- p, n-q(F*) = {I/I

I0'" jl = 0, o( # 1/1)1 = O}.

So

and

#: ytn-p,n-q(F) _

ytp,q(F),

Lemma 4.1 implies that # is a (conjugate linear) isomorphism.

EXAMPLE.

Q.E.D.

Let M be a compact Riemann surface. Then HI(M, l!i)

= HI(M, no) ~ HO(M, nl)

and HO(M, QI) is the space of holomorphic differentials

on M. Thus, dim HI(M, l!i)

= dim

HO(M, nl)

= genus of M,

Further HI(M, 0) = HI(M, l!i(T» ~ HO(M, QI(T*» = HOeM, l!i(T*

® T*»

which is the space of holomorphic quadratic differentials on M. Thus, dim HI(M, 0)

=

(

0, 1, 39 - 3,

if genus (M) if genus (M) if genus (M)

=0 = ~

1 2.

[For example, see Teichmiiller (1940).]

5.

Covariant Differentiation on Kahler Manifolds

In this section we want to exhibit some of the special facts that a Kahlerian structure imposes on the Hermitian geometry of M, a complex manifold. For instance, 0 = 0 = 1-.6. holds on a Kahler manifold. First we must review the idea of a covariant derivative. Suppose ~j(z)(%zj) = ~k(z)(a/az(k) is a given COO section of T. If Z. E V j n Vj n V k =F rjJ, then in general,

La

La

5.

COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS

107

because

and thus,

iP"/:

_'oj _ "

aZ~

-

oz": aJ:

JJ _ J _'ok

L. oZ~ oZ~

jj2 Z ": + " __ J_ ~(J

L.

oZ~ oZ~

k'

We would like to define a "correction" term rj;.p(z) such that

where

We temporarily fix the following notation: will be written

e

YJ

sections in

~(T), ~(T*)

~('f), ~(r*)

L~j(O~j)'

LCPj«dzj,

L'7~(o~j)' Lt/lj«dz~,

respectively. We suppose we have fixed an Hermitian metric ds 2 = L gM dzj /\ dz~ on M. Later we will assume that this metric is Kahler. Let U be a coordinate patch with coordinates (Zl, "', z"). Let 0). = oloz).., 0).. = iJloz).. The '1~ transform as follows:

Since

(::D

o).'7f(z).

V)..'7~ = G.. Ilj , V.. t/ljii =

o.. t/lju,.

o)..'7j(z) =

~

Similarly for t/I j«' We define Let (g~«) = (g MJ) -I. Then p(~) =

II

L giYii~j dz ll E ~(f*)

y= I

(1)

108

GEOMETRY OF COMPLEX MANIFOLDS

and we define

that is,

Thus,

and r~y =

L g~" OAgyp P

=(OAG)'G- 1 , where G = (9 ..p). Similarly,

n

=

L n. lpjy ,

0Alpj.. -

y= 1

since '" 9 gPY = {)Y.. ' ~ ..p p and

0= L OAg .. pg PY p

+ L,g..p OAg Py . p

We also notice that if (Zl, .. " zn) are different coordinates on

a ) OZ"A( a ) (OZA = ~ oi OZ" ' so

Suppose now that

Zj

=

z. Then we see that VA~" dz A ®

(o~. )E !1J(T* ® T)

and VAlp.. dz A ® dz" E ~(T* ® T*),

5.

COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS

109

that is,

.. _ " oZ~ ozj

VjA~j

L, ~;p It,fl uZ J uZk

-

fl

Vklt~k'

This remark is used in differential geometry as motivation for defining V : !5}(E) -+ !i}(T* ® E), where E is a C«> vector bundle over M a differentiable manifold. We also define V, VA~"

= 17}.~", VA({).. = °A ({).. ,

V}.rti = G}.rti

+

L r~flrtll, fl

VAI/Ii = 17 A I/I« -

Lfl r~.. I/Ill·

In fact, we could define V). ({) analogously for any tensor field !5}(T ®

(() E

'" ® T ® ... T* ® ... T*),

by raising or lowering indices until p«({)) E !5}(T ® '" T*) taking 0). and then p -1. We will not write out the result in local coordinates here.

V}. g"l1

Proof.

= L9. ~aA(g~YgYll) =

Lg..~OA(b~) Q.E.D

=0. THEOREM

5.1.

w= i

L 9 ,,11 dz" "

dz ll is Kahler if and only if

r~fl =

Proof.

w is Kahler if and only if dw = 0 and

dw = i

Now r~fl =

f'Py.

L iz«f dz}. "

dz" " dz ll

La g~" oigfl~) implies OAgfl~ = L. 9..~ nfl' and so on. Therefore, i dw = 2L gtll(r!A - rI,,) dz}. dz" " dz ll 1\

GEOMETRY OF COMPLEX MANIFOLDS

110

Thus, dw = 0 if and only if r~1I

=

r py .

PROPOSITION 5.2. Let cp = I/p!q! Kahler manifold, then

L CPI1.I"'l1. pPI

... /lq dzl1.l,. .. ·Adz/lq• If M is a

We prove the case q = O.

Proof.

1 ~ V cp - ,L.. 11. "'I

p.

Q.E.D.

... ",p

d zl1.

A

d Z I1. ' ... d zl1.p

Since r:tl.J = r;JtI. and dztl. A ••• A dz"'p is skew-symmetric all terms sum to , zero except the first which sums to ocp. Q.E.D. Similarly: PROPOSITION 5.3. In the Kahler case, acp -- L.. ~ V11. cp"'I···"p/l,···/I. dz"

THEOREM 5.2.

Proof.

A

dZ""

A ••• A

dz/l q.

In the Kahler case

By Proposition 2.3 -( -l)P(.9cp)
t

(all

+ ~ 011 9 )cpii"

.iipllll, "'11 •.

By definition

L Vll cpAplllll'··II. = L 0flcpApllll,···fI. + L II

II

rSycpAp/l 1 ···flp

7.11

+ L rg~cpApIl7f12 "'11. + ... 7.11

=

L all cpApllfl' '··11. + L r~ycpApfl' ... /lp. II

7.11

5.

cov ARIANT DIFFERENTIATION ON KAHLER MANIFOLDS

q; q;

This is true because = (PPI ... y ••• Pq ). We claim

111

and cpApPPI"'Y"'P o is skew-symmetric in

where g

= det(g",p).

(2)

If we know (2), then

LVPcpApllPI ... Po = L (all + ~ all g)cpif pP"'11. P

g

P

(3)

= -( -l)P+ I (9cp)ifpP ... Po

by Proposition 2.3. We remark that VA~«II =

(VAe)«p,

since

and

thus

v

A

e«lI=

L g«yg 6PV A.e ylJ'

Similarly we may prove that (3) implies -(-l)P L.... ~ VP cpPApB.- (8cp) ApBo' P

so -( -l)p

L g/JY VyCPAp/JPI'"

/J.y

/J o = (8cp)A p8 o '

Thus we prove (2). Recall

qp =

L gil'" OA.(gpa). a

Let A).v be the cofactor of gAv' Then

But

so

og vA. -=gg. iJg A. v

(4)

GEOMETRY OF COMPLEX MANIFOLDS

112

Using (4) and the fact that M is Kahler

Q.E.D. We now introduce an operator A mapping nAP,q) into nAP-l,q-l). Let 1 '" .. , ... "1'/1, "'1/, dz'" cp - -, -,~CP p.q.

A

••• A

dzl/, .

Then 1

igl/llcp dz'" A cp -- (p _ 1)! (q - I)! '" ~ ..1/"2 ... "1'1/2 ... 1/,

A ••• A

dzl/,

'

where i = ~; that is,

If p -1 < 0 or q - 1 < 0 then we set nAP-l,q-l)

= O.

LEMMA 5.1. A is a real operator; that is, Acp = Aip.

Proof. Obvious. PROPOSITION 5.4.

iJA - AiJ = i8, oA - Ao = - i9.

Proof. The first implies the second. We shall prove the first, by Proposition 5.2, (iJAcp) .. , ... "pB = V.. ,(Acp)"2"· .. pB - V...(Acp)"'"J''' "pB + "',

where B =

P2 .. , pq • Thus,

(iJAcp)", ... "pB = i( _1)p-1

L {V",gl/ ..cp....

2 ...

B -

V" 2 gP..cp .... ,.. J ... B + ... }

",/I

-Z·(-I)p-l"'gP .. {V 111 cp 11(12 "'B--V (12 cp 11(1,(13-·· B-+"'} • ~ ... /1

5.

COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS

113

Again by Proposition 5.2

and

Thus,

by Theorem 5.2.

Q.E.D.

THEOREM 5.3. On a Kahler manifold, Proof (1) We prove 0

fj. =

20 = 20.

= O.

0= 08 + 80 = -i{o(oA - Ao) + (oA - Ao)o} = -i{ooA - oAo

+ oAo - Aoo}

D = 09 - 90 = i{ ooA - oAo + oAo - Aoo} = -i{ooA + oAo - oAo - Aoo}. Thus D= O· (2) Proof of fj. = 20 = 2IJ. fj.

= do + od = = =

+ 0)(8 + 9) + (8 + 9)(0 + a) 09 + 90 + 08 + 80 + 08 + 80 + 09 + 9.1 D + 0 + 08 + 80 + 09 + 90. (0

But i(08

+ 80) = o(oA -

Ao) + (oA - Ao)o = 0

so 08 + 80 = 0, 09

+ 90 = o.

Thus, fj. =

20 = 20.

Q.E.D.

Now we wish to derive the important relations between Hr(M, C) and Hq(M,OP).

GEOMETRY OF COMPLEX MANIFOLDS

114 THEOREM 5.4. M we have (1) (2)

(Hodge, Kodaira, deRham) On a compact Kahler manifold

Hq(M, OP) ~ HP(M, oq), H'(M, C) ~ EB HP(M, oq). p+q=r

Proof

(I)

Hq(M, OP)

~

Jfp,q = {q> I q>

HP(M, oq)

~

Jfq,p

E

r(AP,q), Dq> = O}

= {q> I q> E r(Aq,P), Dq> = O}.

The map q> --. iP is an antilinear isomorphism from r(Ap.q) to f(AM). If Dq> = 0, then CiiP = o. But M is Kahler so D = D, and conjugation thus gives an isomorphism Jfp.q ~ Jfq,p. (2) By the de Rham theorem,

But Hr(M, C)

~

Jfr = {q> I q>

E

r(A r ,6.q> = O}.

(5)

This is the Hodge theorem, We have not proved it, but its proof is similar to that of Theorem 4.1, using de Rham's theorem and the decomposition .!l'q = Jfq + d.!l'q-l + ().!l'q-l analogous to that of Proposition 4.2. We leave it as an exercise to the reader. We have the following decomposition:

= EB [(AM).

r(Ar)

p+q=r

We claim Jfr

= EB Jfp,q. p+q=r

(6)

For if 6.q> = 0, then Dq> = t6.q> = O. D maps (p, q)-forms into (p, q)-forms and thus Dq> = 0 implies Dq>(p,q) = 0, where q> E r(Ar) and q> = Lp+q=r q>(p,q), q>(p,q) E r(AP,q). Thus 6.q> = 0 implies qJ(p,q) E Jfp,q. So Jfr

£;

EB Jfp,q. p+q

The reverse inclusion is also easy and (6) is proved. This proves Theorem 5.4. (2). Q.E.D. Let hp,q = hp,q(M) = dim Hq(M, OP) and br = br(M) = dim Hr(M, C); br is the rth Betti number of M. COROLLARY I.

On a compact Kahler manifold b 2p

~

hP'P

~

1.

5.

COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS

Proof h 2p ~ (p-times). Then

hP'P

is clear. Let w

= i L gapdz a " dz P w P = w

115

" .. , " w

Dw P =0,

because w P = (i)P '\' - ... dZrJ. 1 " dZ iil ., • L... 9 a,P- I 9 a2(!,

and V), gaP = 0,

V), gaP =0.

So, ow P = 0, 9w P = 0, by Proposition 5.3 and Theorem 5.2. Thus, Dw P = O. From the calculations in Theorem 1.4, we know w P =1= O. Thus dim yt'P,P > O. Q.E.D. From Theorem 1.4 we already know h2p

REMARK.

COROLLARY 2.

~

I.

On a compact Kahler manifold b2 k+ 1 == 0 (mod 2)

Q.E.D.

Proof

PROPOSITION 5.5.

On a compact Kahler manifold, every holomorphic = O.

p-form

Proof

d
= o

Thus,

I o

= (o
oA - Ao = II d

= -

- i

9 so

i (
o

EXAM?LE. We show that Kahler is needed in this proposition. Let be the subgroup of GL(3, q defined by

C'~{(Z"Z"Z')]~(ZIZ~G

l :m"GL3 i

,C)

1[3

116

GEOMETRY OF COMPLEX MANIFOLDS

Let

G+ (~ l' i:) =

Ig,= m, + in,; m"

n,ell

One easily checks that G is a subgroup of C 3 • The quotient space C 3 /G is a compact complex manifold. For

1 zl+91 1 o 0

Z9 = ( 0

Z2+92+93 Z,) Z3

+ 93

I

= Z ,

1

1 z~ and each point of M is represented by some Z' = ( 0 1 o 1

Z;) withz;' = x;' +

z; 1

i y~ where 0 $ x;' $ I, 0 $ y;' $ 1. The terms of Z' satisfy

z; =

ZI

+ 9"

z~ = Z2

+ 92 + 93 ZI' z;

= Z3

+ 93'

Thus

=

dZ 2

+ (Z3 - Z3) dz,.

Hence, cp = dz; - Z3dz; = dZ 2 - Z3dzl is a holomorphic I-form on C 3 invariant under G; if 1[ : C 3 -+ C 3 /G = M is the canonical (holomorphic) map, there is a well-defined holomorphic I-form 1/1 on M such that 1[*(1/1) = cpo Similarly there are nonzero forms ~, '1 on M such that 1[*(~) = dz 1 , 1[*('1) = dZ 2 Since dZ 3 /\ dZ I =F 0, ~

/\ '1 =F O.

But dcp = dZ 3 /\ dZ I and hence

dl/l = ~ /\

'1 =F O.

Thus the theorem is not true in general

6.

Curvatures on Kahler Manifolds

We assume throughout this section that M is compact and Kahler: We wish to find some expressions for the Laplacian in terms of curvature tensors which will be used in Section 7 in the proof of the vanishing theorems. Let co = i L 9,.p dz" /\ dz P be the Kahler form on the Kahler manifold M. We define the bracket of VA and V•.

6. PROPOSITION

[V A'

6.1.

VyJe« =

III = 1 RpiiA ep, where

-

RpiiA = DEFINITION

117

CURVATURES ON KAHLER MANIFOLDS

0. TAp .

The tensor field Rim is called the curvature of the metric w.

6.1.

(of the proposition).

Proof

\lye« = Oye«, so

Similarly, \ly(VAe«) = oY(OAe« =

+ ~ r~p)ep

0. 0Ae« + I TAp o.e' + I o.TApe'. p

II

Thus Q.E.D.

I:=

1 g/Jii. R~iiA' We wish to investigate the symmetries of Rp'IA' Let Rii.PiiA = By Theorem 1.2 we can find real COO Ku on each small coordinate patch U eM such that g«p = o«of/KU on U. We claim:

PROPOSITION

6.2. n

Rif/iiA

= a«of/o.oAK u Il,

L T=

gii'
on any small open set where Ku is a Coo real function.

Then

L glt.RpiiA = L g«.O.giilto;.gf/ii + O.OAg,.· It

Since

/J.1t

GEOMETRY OF COMPLEX MANIFOLDS

118

Thus

"',,, =

a. 0;. at op Ku -

L g#"'(o", ato. Ku)(a" op 0;. Ku)·

Q.E.D.

"',/l

(1) R«p.;. = R.p«;. = R.;.«p = RrmfJ .

PROPOSITION 6.3.

(2)

R«fJV). = Rl1czX./l •

Proof Proposition 6.3 (1) is clear from Proposition 6.2. For part (2), conjugate 6.2 and remember that Ku is real. Q.E.D.

DEFINITION 6.2.

The tensor field

is called the Ricci curvature. PROPOSITION 6.4. Proof

R.;.

= OAO. log g, where 9 = det(g",l1)'

0. LfJ r1fJ = 0.0;. log 9 by (2), Section 5 of this chapter.

R.;. =

We have a few more simple computational results.

Proof

V. ({J", =

0. ({J"" so

V;. V.({J", = 0;.0.({J1JI -

LfJ r1",0.({Jp.

Since we also have V;.({J", = o;.({J1JI -

" r1",({Jp, L fJ=l

we get,

V. V;.({J", = o.(V;.({J",) = a. 0;. ({J", -

Lp o.r1",({Jp - L r~",o.({Jp. p

Thus Q.E.D.

Q.E.D.

CURVATURES ON KAHLER MANIFOLDS

6.

PROPOSITION 6.6. Proof

119

- .]lPa = - " [V A' V ~p p Ra. vA lPp·

Conjugate Proposition 6.5 to get [VA' V.JqJlZ =

L R!iiAipp

Q.E.D.

We could similarly prove [VA' V.J~IZIIi' = -

L R~A' ~tlli' + L RpAy ~lZtji t

+

t

L Rhy~lZlIi' t

THEOREM 6.1.

For any (p, q)-form 1P = Ijp!q!

(OlP )IZI ···P.

=-

" ... " dz Pq .

L gPIZVIZ V1I1P1Z' ... /I.

IZ,P

P

+

L 1P1Z1'''P. dZIZ'

q

L L t.a L RtlZi/lkiilPlZl"'lZi_ttIZO+t"'/lk_tii/lk~t···P. i=1 k=1

q

- L L R/lki 1PIZt ... Pk-tiPk~ k= I

Proof

t •••

/I.'

t

As usual let A denote oc i

.••

a p • Then

(OlP)..tPo" P. = (-1)P{V po lPAPtP2'" - V llt lPA/loP2'"

+ ... }

and (90lP)AP"

P. = -( -l)P = -

L gPIZV,.{OlP)A/lPI ···P.

lZ.fJ

L (gPIZVIZVfJlPAPt .. P.+ A=t

IZ.P

(_l)HlgPIZVIZVp;.cpA/I .. i;. ... P.). I

120

GEOMETRY OF COMPLEX MANIFOLDS

Also

so

Thus (0<,0 )A/I, ... /I. = (80 = -

+ 08)<,0 A/I, ... /I.

L"

a.p= 1

gIIa.Va VP<'oA/I, ... /1.

(1)

q

- L (_l)).g/la[va , V/1,,]<,0 A/I··· P... ... /I.' ).=

1

Let us calculate the second term on the right-hand side of (1). For a form <,0 of type (l, 0)

L Rta;;). <'ot

[V)., V,]<'oa =

t

and ,,/I). _" L.. g [V)., V,]<'oa - L..

R t a;; P<'ot'

).

For a form <,0 of type (0, 1) [V)., V,]<'o/I=

-I Rp'ii).<'o, t

so

We also have

L Rp'p,P = L gragPYR/la/l,y /I Ct. P. = L g'Ct.R Pla = R/I," Ct. Y

Similarly, we see

L gPCt.[VCt.' Vp,,]<'o,4/Jtj, ... P. . ·.. P.

a./I

(2)

6.

121

CURVATURES ON KAHLER MANIFOLDS

Since RSkiS/ is symmetric in is zero. Thus, using (2)

fP and

cP is antisymmetric in

fP,

the last term (3)

where (t)i means that t occurs in the ith place. Multiplying (2) by (-1»). and Q.E.D. plugging into (I) yields the theorem. We want to derive a similar theorem for Do acting on r(Ap.q(F» where F is a complex line bundle defined by the I-cocycle {./jk}. (As usual in this section we are assuming that M is Kahler.) A form cP E r(AM(F» is given locally by a family of (p, q)-forms {cpj} on {Uj} where {Uj} is a covering of M with coordinate patches over which F is trivial such that

CPj =./jkCPk on Uj n Uk.

(4)

Let en = i 2: g,.s dz" 1\ dz S be the Kahler form of M and suppose we have chosen an Hermitian form (,) on the fibres, so

(C,O = ajlCjl2, where Cj is a fibre coordinate of Cand aiz) is a real positive Coo function on U j Then aj IC j l2 = ak ICkl 2 implies

I/j kl 2 = ~.

(5)

aj

For two forms cP, I/J E r(AM(F» their inner product (cp, I/J) is then

(cp, I/J) =

f ajcpj M

1\

* i/ij.

The integrand is well defined since by (4) and (5),

ajcpj

1\

* ili j =

akCPk

1\

* i/ik on

Uj n Uk.

Recall that we defined

and by Equation (1), Section 3

* (aj1aaj 1\ * CPj) 8cpj - * (2: aj la,.a j dz" 1\ * CPJ - * (a * cp) - * (2: aj 1 0,. aj dz" 1\ * CPj).

(8 a cp)j = 8cpj = =

•

122

GEOMETRY OF COMPLEX MANIFOLDS

Recall also by Proposition 5.2 and Theorem 5.2 9cpj = -

* (0 * cp) =

-

*

Ct

Va dz a "

* cPj)

and

Thus we have: PROPOSITION

6.7.

(9,.CP)a, oooJ/, oo0{Jq_, = -( -lY L gJ/a(Va a,{J

Proof

+ Oa log aj)CPja,.ooapJ/J/,oooJ/q_,.

We need only show

To do this we assume glla(zo)

= {)lla and CPj = dz Ap

dz llq . Then

"

* CPj = sgn(Bq Bn- q )in( _l)tn(n-l)+n p dz Bn - q " ApAn_p

dz An - p •

Next

* (dz a "

dz Bn - q

"

dz An - p) = Yfdz Ap

"

dz x,

where '1 and X are as follows: X is the increasing set of numbers (Xl··· X q - 1 ) complementary to the set a.Bn - q £ (I .. ·n). If we order a.Bn _ q in increasing order Y let e = sgn

(BY). a. n-q

Then '1

= ein( _l)tn(n- l)+n(n-q+ 1) Sgll = .n(_l)tn(n-l)+n(n- q + 1) I

Thus,

Hence,

sgn

(-;- P

~)

(An_pAp) B X· a. n-q

6.

123

CURVATURES ON KAHLER MANIFOLDS

where X = (Xl' •. Xq-l) is an increasing set of numbers from (I ... n) so that X is the complement of IX in Bq • But we also have

L

(-l)P IX

cltlpjApax,,,,xq_,

dz Ap

1\

dz x

v X=Bq

= (-l)P

L

Ole Bq

COl

sgn (B q ) dz Ap IXX

1\

dz X

corresponding to the right-hand side of (6). Letting Cit = ajlo"aj we see (6) is verified at Zo in the case gI1«(zo) = {)111t, lpj = dz Ap 1\ dz fJ •• The general case Q.E.D. follows easily from this. We want to define covariant differentiation of sections lp E r(AP·q(F». Let lp = {lp J E r(A(p,q)(F». Then lp jAI1 = fjk lpkAI1' One can easily check the following fact: If lp is a form and f is a C" function,

and VIt(f' lp) = (o,J)lp

But in our case

Ofjk

+ f· Vltlp·

= 0 so

Thus we define (7)

However,

v

It

lp lAB = !lk VIt lpkAB

+ Olt!lk lpkAB

so we must make a different definition ofVIt which depends on

ak aj

2

J

- = I!jk I = !jk jk'

Hence,

Now

so that

aj'

We know

124

GEOMETRY OF COMPLEX MANIFOLDS

thus proving

We define

V~tJlqJ =

{:j v. (a jqJ

(8)

j )}

= {(V.. + 0.. log aj)(qJj)}'

THEOREM

6.2. III O tJ"f'jA,B. - -

gll"V(tJlV IfI .. p."jA,B,

'" ~

q

+ k=L1 L(X i llk - R llki )qJjApll.···(t)k···II. f

X

Li Lk L Rf"llIkjj qJ

j .. , ..• (ill ... "I'll,

... Cjjlk ",11.'

t,i

where

X/ = - VPe and e = L.. g'''o.. log aj •

Proof

We omit the subscriptj from our calculations. From Proposition

6.7, (8tJ qJ)ApB.-, = (8qJ)A pB4 _,

-

(

-1)P

LP ellqJApIlB.-t •

We denote the last term in this expression by ('qJ)ApB._ t • Then

+ 8tJ O)qJ]A pB. = (OqJ)ApB. + [(oe + eO)qJ]ApB,.

(OaqJ)ApB, = [(08tJ

Computing gives (using Proposition 5.3),

__

-

~

~(

_

Ie=1 q

= -

We also have

k-l-

1)

~

II

VPk ~ e qJApllllt .. · 8k'" II. P

L LVPk(ePqJA pll,···
k= 1 P

7.

VANISHING THEOREMS

125

So

We now use Theorem 6.1. First note that

- L g/la;Va; fill -

~flVIl

L gfla;(Va; + oa; log OJ)'lp - L gfla;v~a)v p •

==

Now use (9), (10), and Theorem 6.1 to finish the proof. REMARK.

Q.E.D.

L gA,X'ji = -VjiOA log oJ 04°;; log OJ =

XAji =

(10)

-°

(ll)

is called the .curvature of the metric a.

7.

Vanishing Theorems

We wish to use the computations of Section 6 to show that the cohomology groups H9(M, ~(F» must vanish under certain circumstances. The first result to this effect is the following theorem. The technique is due to Bochner. THEOREM 7.1. If the Hermitian matrix X ta - Rat is positive definite at each point of the Kahler manifold M, then

H9(M,

~(F»

= 0

for q ~ 1.

Proof We first prove: LEMMA 7.1. Let ro = i L 9a;/I dz" any I-form on M. Then

A

dz fl be the Kahler form on M. Let be

Proof qJ A *~ = (qJ, t/I)(z)(ron/n!) from Theorem 2.1. Let qJ = 1, ~ = 1 be a differentiable function. Then we see that *1 = I(z)(ron/n!). Recall that JM d'P = 0 for any (2n - I)-form 'P. Thus,

0=

t

d(*
t

± **d(*
=

t

± *.5<1> = ± f.5<1>· (ron/n!).

Q.E.D.

126

GEOMETRY OF COMPLEX MANIFOLDS Now for the proof of the theorem. Recall that

Hq(M,I!l(F»

~

,1fo,Q(F) = {IP lIP E f(Ao,q(F», DalP = O}.

Thus we want to show that any IP sarily zero. Let

E

f(AO,q(F» satisfying DalP = 0 is neces-

Then is a (0, I)-form so OJ"

OJ"

O=f c5<1>·-=f .9<1>-M

n!

M

n!

since

is the zero map. Thus,

f

= aj{I giJ~ I

+

v~a)v{lIP jBq • IPfq} :~

f AI glJ~ I V{lIP a

V~ IPfq} :~ .

jBq

The term in the braces in the last term is always nonnegative since g~lJ and glJ~ are positive definite. Thus the second integral is nonnegative and the first integral is nonpositive. It is clear from the derivation of Equation (3) in Section 6 that for a (0, q)-form IP using Theorem 6.2

Thus,

since Da IP j = O. Recalling the definition of lP~q we see

0> -

fa

j

q A, -

" L It

Bq _ I

"(X L ta - R)g at ~,lJ, "'g ~q-,lJq-, CT I

,. t n .

By assumption (X ta - Rat) is positive definite. Hence, ,1f(o,Q)(F) = 0 for q ~ I. Q.E.D. REMARK.

- - OJ"

cptA q -, • .." m aBq -'

For q = 0 the theorem says nothing.

cpt~''''~q-, =

0 and

7.

127

VANISHING THEOREMS

We now discuss the meaning of the curvature XJ.ii. Recall the sequence

o-----. 71. -----. (!) -----. (!)* -----. 0 in Section 5, Chapter 2. We get the exact cohomology sequence ... -----.H1(M,

(!) -----.

Hl(M,

6"

(!)*)-----.

H2(M, 71.)--+ ...

and we defined c(F) = ch(F). Since 71. c C we map H2(M, 71.) send c(F) -+ c(F)c. De Rham's theorem then says

-+

H2(M, C) and

r(dA I) H2(M C)~--. , - dr(A 1 ) THEOREM 7.2. The de Rham cohomology class of c(F)c is represented by (I/2ni) Xlii dzJ. A dz V•

L

Proo}: This is an exercise in tracing the de Rham isomorphism map. Let F = {fjk}. Then c(F) = [{cijdJ, where Cijk

We wish to find')' U j such that

E

1

= 2-----: {logfij nl

+ logfjk + logfki}·

r(dAl) representing c(F)c. We can find Coo I-forms u j on I - . d 10g!jk 2m

= Uk - u j

Then,), = dU j = dUk. Remember that l!jkl 2 Thus

log!jk

+ log!jk =

•

= ak/aj

and X;'ii

= -oj. oil.

log ak - log aj

and

Let

Then ')' =

1", 1,3 AdUk = - . () iJ log ak = - - . L.. oJ.u/J.log a k dz A dz/J. 2nl

2nl

Q.E.D.

log aj .

128

GEOMETRY OF COMPLEX MANIFOLDS The line bundle over M defined by the l-cocycle a(z~

K

... zZ)

= {J jk }, Jj/c = at Zj1 ••• Zj")'

with respect to a coordinate covering {U j} of M, is called the canonical bundle K of M. Then it is easy to see that

9j = IJ jk 12 •

9k The first Chern class c1(M) of M is then (this may be taken as a definition) C1(M)

THEOREM 7.3.

= - c(K).

The de Rham cohomology class of c1(M)c is

[~ L RilA dz A 2111

Proof.

1\

dz il ].

Left to the reader.

THEOREM 7.4. Let M be a Kahler manifold. Let F = {file} be a line bundle. If a (I, I)-form 1 L YA - dz A 1\ dzP.Y = -. 2111 P.

is real (that is, y = Y), dy = 0, and if [yJ = c(F), then there exists {aj}' a j Coo functions on U j ' aj > 0 satisfying aj \.fj1e\2 = ale such that

i

y= 21l a0 log a

Proof. and

a l.fjkl j

2

j'

Choose any metric il = {aj} on F. That is, il j = k • Then define

a

1 . L X Ail dz A 1\ dz il , ~ =-2111 where that is,

i ~ = 2n:

a0 log a

j .

E

Coo(U j ), ti j > 0

7. Then as in Theorem 7.2 dcp is a (1, I)-form and

VANISHING THEOREMS

[~]

129

= c(F)c so ~ -

y = dcp, where cp is a I-form. Thus,

dcp = ~ - y = 11

+ Otjl,

where 11 and tjI are (I, I)-forms and 011 = O. But then

A11 = 2 011 = 0

d11

so

= t511 = o.

Also ddcp = 0 so dcp = 11

+ !(dt5 + t5d) tjI

implies 0= dt5dtjl

(1)

and hence (t5dtjl, t5dtjl) = (dtjl, dt5dtjl) = O.

Thus, so Then (11, dcp) = (t511, cp) = 0

= (11, 11) + ! (11, dt5tj1) = (11, t7). Hence t7 = o. Using Equation (J) 0= (t5dtjl, tjI) = (dtjl, dtjl)

so dtjl=O.

(2)

From (2)

o=

dtjl = atjl

+ atjl.

Thus, atjl = atjl = 0 since tjI is of type (I, 1). So dcp =

e- y = a8tj1 = -i(aAatjl- a aAtjI) = i a oAtjI.

Thus, i 00/, 211:

~ -y=-

130

GEOMETRY OF COMPLEX MANIFOLDS

where I is a Coo function on M. But

e- y = e- - y = -

; ~ 2n u oj

i

= 2n 0 oj

e-

e-

since y is a real form. Hence, y = (ij2n)0 vU)(1 + j) and thus we may assume that y = (ij2n)o 0I where I is real valued. Finally

e-

Y=

i_ e- 2n -; 0 of_ = 2n - 0 o(log a. - f). J

i

(1) Y = - 0 0 log a . 2n J

Q.E.D.

REMARK.

Perhaps we should explain this proof a little more clearly. We

claim: PROPOSITION

7.1.

If 1

1/1 = -2. I l/Ia11 dz" nl

1\

dz 11

and jf [1/1] = 0 (that is, 1/1 = dcp), then there is a Coo function 1/1 = 0 01 when M is a Kahler manifold.

!')

I such that

Proof Let 'P={I/III/I=dcp, 1/1 of type (I,I)}. Then 00!,)~'P, where is the space of differentiable functions, and 'P = YEt> 0 0 f!}, where

Y

= {'71 '7 e 'P, ('7, 0(/) = 0,

for all/e f!}}.

We note that ('7, 0 0f) = 0 if and only if 88 '7 = O. We claim that if M is Kahler, then Y = {O}. For Kahler implies t~ = 0 = 0 and 08 + 80 = 0 = 08 + 90. Thus t~2'7 = 00'7 = (v8 + 80)(09 + 80)'7 for '7 e Y. Since '7 is of type (I, I) and '7 = dcp, 0 = d'7 = 0'7 = 0'7. Thus, t~2'7 = (0809

+ 8009)'7

= - 0089 '7 + 8080 '7 =0. Thus t1 2 '7 = 0 and (~2'7, '7) = (t1'7, ~'7) = O. So t1'7 = 0 and hence 15'7 Finally. ('7, '7) = (dcp, '7) = (cp, 15'7) = 0 and '7 = O. Q.E.D.

=0

7.

131

VANISHING THEOREMS

DEFINITION 7.1. A complex line bundle F over any compact complex manifold is said to be positive if there is a y = (l/2ni) X).il dz). /\ dz il , dy = 0, y = y, and [y] = c(F)c such that X).il(z) is positive definite at every point z of M.

L

L

REMARK. If F over M is positive, then w = i X).jj dz). /\ dz jj is a Kahler form. Hence M is a Kahler manifold. Rewording Theorem 7.1 gives: THEOREM 7.S.

If F - K is positive, then Hq(M, l!J(F» = 0 for q ~ I.

THEOREM 7.6.

If -F is positive, then Hq(M, l!J(F» = 0 for q :::;; n - 1.

Proof Serre duality gives Hq(M, OP(F» ~ Hn-q(M, on-pc -F» where dim M = n. Notice that on ~ l!J(K) and let p = O. Then

Hq(M, OO(F»

~

Hn-q(M, on( -F»

~

Hn-q(M, l!J(K - F»

= 0 fO[ n - q ~ if K - F - K

=

-F is positive.

1

Q.E.D.

We also have: THEOREM 7.7. If F is "sufficiently" positive, then Hq(M, OP(F» = 0 (where F is a line bundle) for q ~ I.

Proof

Again we use

Hq(M, OP(F»

~

.yt>P·q(F)

= {cp I DaCP

For cP

-I

0>

M

E

= 0, cP of type (p, q)}.

ytJ(P, q)(F) we let the reader check the following inequality:

,q { L L (X ij n. t.1' «,.P"

Wn

t

_

t

_

Rij )CPl1.i ... l1. p tP2 ••. /I. cP

ii, ...

iiptllh·· P.

Thus if X'ij is sufficiently positive definite, then the integrand is positive for cP =1= 0 we see ytJp.q = O. Q.E.D. We now proceed to a generalization of Theorem 7.6 due to Nakano (1955). As usual M is a compact Kahler manifold and F = {.fjk} is a complex

GEOMETRY OF COMPLEX MANIFOLDS

132

line bundle with metric {aj}. Remember that (8"cp)j = {(l/a)8(ajcpj)}, and so forth.

(00" + o"o)cp = X /\ cP, where X = -00 log aj and

LEMMA 7.2.

r

cP

E

(AP·q(F)).

Proof

We have (o"cp)j

=

{~j o(aj' CP)}

= {ocpj

+ alog aj /\ CPj}.

Thus,

O(o"cp)j={Oocpj+oologaj /\ cp-ologaj /\ oCPj}. Add o"oCPj

= {oocpj + alogaj /\ oCPj}

to get (00"

+ o"o)cp = X

/\ cpo Q.E.D.

THEOREM 7.8. [Nakano (1955); Calabi and Vesentini (1960)] r (AP·q(F» be such that ocp = 8" cP = O. Then

Let cP E

0::::;; )=l(X /\ Acp - A(X /\ cp), cp). Proof since

0::::;;

FI (o"cp, o"cp) = (90"cp, cp)

(01/1,1/1) = (cp, 8" 1/1) and (9cp, 1/1) = (cp, Ao. Hence

-FI9 = oA -

0::::;; (90"cp, cp) =

FI (oAo"cp -

a" 1/1).

By

Proposition

Aoo"cp, cp)

5.4

(3)

=)=1 (Ao"cp, 8"cp) - )=l(A(oo" + o/)cp, cp) = -J=l (A (X /\ cp), cp). But we also have

0::::;; (9cp, 9cp)

since ocp = 8"cp

= O.

= (0,,9cp, cp) = )-~I (olJAcp - o"Aocp, cp) = )-=-1 (o"oAcp, cp) = )-=-1 (a" 0Arp + oo"Acp, cp)

Thus,

)-=-1 (X /\

Acp, cp) ~

O.

Now as Equations (3) and (4) to get the theorem. THEOREM 7.9.

[Nakano (1955)]

Q.E.D.

If F is negative, then

Hq(M, QP(F») = 0 when n = dim M.

(4)

for p

+ q ::::;; n -

I

7. Proof

133

VANISHING THEOREMS

By the harmonic theory

Hq(M, QP(F»

~ {q> I q> E

P,q(F), aq> = 911. q> = O} = Jf('p.q(F),

where P,q(F) = r(Ap.q(F». By Theorem 7.8 if q>

o :s; ~l(X

E

Jlfp,q(F)

" Aq> - A(X " q», q».

We let the reader verify the following equations: (X " III) 'F 11.0 ... «q/lo ... /lq = (-l)P '" L... X «0/10 III 'F«,

•..

«p/l, ... /lq

P

+

L(-lix«./loq>«o ... a, ... /I, ...

,=1 q

+ k=l L(-ltX«./lkq>«\···«p/l.··.Pk ..

/lq

P

~

~

L

L ( _l)kgjiAX A/lk q>«, ... ji/l • ... Pk"

i;iA + /.t,L...A i=L...(-l)g X«,jiq>A« •... ri.···/I\/ll··· 1 q

+

/l,A k= 1

p,q

+

'L... " 'L... " ( /l,). i=l,k=l

-1) iHX Cli/l. 9 jiA q> A··· ji ....

Thus,

Since F is negative, - X Aji is positive definite at each point. (We should use Theorem 7.4 here; that is, we choose aj so that - X Aji is positive definite.) Now

GEOMETRY OF COMPLEX MANIFOLDS

134 and

00

satisfies

doo = deL -

xA" dz A 1\ dz") = o.

Thus we may use - X A" as a Kahler metric on M. Hence, assume g",P Then

= - X",p .

and

Finally

O$f

M

0$ So 0 $ (-n

1 ,-,-,{(-n)L+PL+qL}, n. p.q.

00"

1 '" ---(-n+p+q)L.,.CPA fMn!p!q! 00"

+ p + q)(cp, cp).

But p

+ q < n and

B

cP TB p q.

P4

we see that cP must be

o. Q.E.D.

8.

Hodge Manifolds

Recall that by de Rham's theorem a Kahler form on a manifold M determines an element of H2(M, C). We also have the image of the canonical map

H2(M, Z) ----+ H2(M, C) which we denote c -+ Cc •

L

DEFINITION 8.1. gall dz a dz P is a Hodge metric on M if [00] = Cc for some c E H2(M, Z) where 00 = i gaP dz a 1\ dz ll . If M has a Hodge metric, then we say that M is a Hodge manifold.

L

THEOREM 8.1. M is a Hodge manifold if and only if there exists a positive line bundle FE HI(M, l!!*).

Proof Suppose FE Hl(M, l!!*) is positive. Then, by Theorem 7.4, c(F) E H2(M, Z) is cohomologous to (\/21[;) X where X = -00 log aj = X A" dz A 1\ dz ii and (X Aii ) is positive definite. Thus gAji = (1/21[)X A" defines a Hodge metric on M.

L

L

Next we assume M has a Hodge metric, that is, W = i gall dz a 1\ dz P with W cohomologous to Cc for C E H2(M, Z), and (gall) positive definite. It

8.

135

HODGE MANIFOLDS

suffices to show that there is a line bundle F such that c(F) = c; because then 1 c( -F) ""' - . 21t1

L X«fJ dz« "

dz lJ ,

with X «fJ = 21t g«fJ' and hence - F is positive. Recall the exact sequence ... _

HI(M, (!)*)---..:..... H2(M, Z)~H2(M, ( ! ) _ ... F-c(F).

Thus, it suffices to show JlC = 0. Let c be defined by the 2-cocyc\e c = {c ijk}' The proof involves chasing through the de Rham and Dolbeault isomorphisms. Consider the following diagram: /H2(M, C) ~ J'(dAl)/dr(AI) CEH2(M,Z)~

cc-t/I

(I)

H2(M, (!) ~ r(oA O• 1 )/or(A o•1 )

/J

JlC _

fP(0.2).

As in the argument of Theorem 7.2, we can find differentiable functions Aij such that Cijk = t5(A};jk = Ajk + Aki + Aij' Then we can find differentiable I-forms t/lj such that dAjk = I/Ik - t/lj. Then t/I in Diagram (1) is obtained by 1/1 = dt/lk = dl/l j . For the Dolbeault isomorphism, OAjk = fPk - fPj' where the fPj are (0, I)-forms. Then fP = OfPk' We can split upl/lj = I/IP' O) + I/IP' O) into forms of type (\,0) and of type (0, 1). We know that d = a+ 0 so we compute ;)1

_ .1.(1.0) 'l'k -

.1.(1.0) 'I' j ,

31

_ .1.(0.1)

.1.(0.1)

UlI.jk -

UlI.jk-'I'k

-'I'j

•

Thus we may assume that fPk = I/Ik(O.1). Then fP = ol/l}O.I) = 1/1(0.2) [the (0,2) part of t/I]. Thus, if Cc +-+ 1/1, JlC +-+ 1/1(0. 2). Now we have assumed Cc '" w which is of type (1,1). Thus 1/1 = W(I.I) + d'1, with '1 = '1 0 •0 ) + '1(0.1). Thus 1/1(0.2) = 0'1(0.1) which means JlC = O. Q.E.D. With the obvious definition of elements of type (\, I) in H2(M, Z) we have: COROLLARY. Let M be a compact complex manifold. Then the il1'age of the map Hl(M, (!)*)~H2(M, Z) is the set of elements of type (I, I).

We now give the proof of the main theorem of this chapter which can be considered as a generalization of the fact that every compact Riemann surface is algebraic.

136

GEOMETRY OF COMPLEX MANIFOLDS

THEOREM 8.2. [Kodaira (1954)] Every Hodge manifold is algebraic (that is it is a submanifold of some pH). We first outline the idea. We know there is a posItIve line bundle

E E H 1(M, £D*). Let F = mE where m is a large positive integer. Let dim HO(M, £D(F» = N

+1

choose a basis {Po, ... ,PN} for HO(M, £D(F», and let F be defined by the I-cocycle {jjk} with respect to some covering {U j } of M (remembering that the jjk are never zero). By definition

fJv = {fJvlz)}, fJVj(z) = jjk(Z) . PVk(Z), where the Pv/z) are holomorphic on U j : M -+ pH given by

•

Consider the candidate for a map for

ZE

Uj

•

It is easy to see this is well defined as a point of pH if for every Z E M there is an index v such that Pv(z) oF O. We want to be an embedding. To prove this it suffices to prove: (1) Given Z E M, at least one fJv(z) oF 0, that is, there is a cp E HO(M, £D(F», cP = CvPv such that lp(z) oF O. [Then (I) implies that is well defined and holomorphic on M.J (2) is injective, that is, for any pair of points p, q E M, there is lp E HO(M, £D(F» such that lp(p) oF 0, cp(q) = o. [In fact, this also implies (I).J (3) is biholomorphic, that is, for each point P there exist n (= dim M) elements lpl' ... , lpn E HO(M, £D(F» such that

L

det where P e U j

(alp~~~Z)) oF 0

•

We first prove (2). Let 1/ = £D(F - P - q) be the subsheaf of (9(F) consisting of germs of holomorphic sections of F which are zero at p and q. Let us investigate the stalks of 1/. Clearly,

1/% = £D(F)% ,

if Z oF p, z oF q I/p={lpE£D(F)p!lpj(p)=O, ;fpeU j }

and similarly for q. We have the exact sequence

o----. 1/ ----. £D( F) ----. 1/" ----. 0,

(2)

where 1/" = £D(F)/I/ is the quotient sheaf. Then 1/; = 0 except at p or q. Clearly 1/; ~ C, ~ C and the isomorphism depends on the choice of local

9';

8.

137

HODGE MANIFOLDS

coordinates around p and q. This shows that HO(M, f/") cohomology sequence of (2) is

= C Ei3 C. The exact

O---+HO(M, f/)---+HO(M, (!J(F»---+C Ef> C---+Hl(M,!J')---+ ... (fJ-«(fJj(P), (fJk(q»·

We sometimes use the suggestive notation f/ = (!)(F - P - q). To prove (2) it is sufficient to prove: PROPOSITION 8.1. If M is compact and Kahler and Fe Hl(M, (!J*) is "sufficiently positive," then

Hl(M, (!)(F - P - q» =

o.

Proof The proof makes use of the quadric transformations Qp, Qq. Let Nt = Qp Qq(M) and let P be the holomorphic map P : Nt -+ M of Nt onto M such that C = p-l(p) and D = p-l(q) are isomorphic to pn-l with dim M = n, and P is a biholomorphic map on Nt - c - D, P : Nt - c - D-+ M - P - q. Let f/ = (!J(F - P - q), IJ = (!J(p - C - D), where P is the holomorphic line bundle on Nt induced by P and IJ is the sheaf of germs of holomorphic sections of P which vanish on C and D. Let dlt = {U j } be a covering of M. Then dii = {OJ}, OJ = P-1(U J) is a covering of Nt. We recall that Hq(M, !J')

= lim Hq(dlt, f/) u

and for q = 1, the map Hl(dlt, f/) -+ Hl(M, f/) is injective. We prove: LEMMA

8.1.

If

then Hl(M, (!J(F - P - q» = O. Proof It suffices to show Hl(dlt, f/) = 0 for all coverings dlt. Take a l-cocycle (fJ = {(fJij} e Hl(dlt, f/), (fJij e rcU i n U j , f/), where (fJij is a holomorphic section of F over U i n Uj such that (fJiip) = 0, (fJij(q) = 0 if p, q e U i n U j • P induces cPij = P*(fJij = (fJij 0 P e rcOi n OJ, (!J(F»,

where cPij vanishes on C and D if C £ Ojn OJ, D £ Oi n OJ. Thus {cPij} represents an element of Hl(dii, IJ) £ Hl(M, fJ) = o. Hence cPij"" 0, that is, cPij = t/Jj- t/Ji where each t/Ji e reUi> (!J(F» and vanishes on C and D. If Ui £ M - P - q, then P : OJ -+ U i is biholomorphic. In this case there is (fJjer(Uj,(!J(F» such that t/Jj=P*«(fJj). If, for instance, peU j, then

138

GEOMETRY OF COMPLEX MANIFOLDS

P: OJ-C-Vj-p is biholomorphic. Hence there is CPjEr(Vj-p,~(F» such that P*(CPi) = '" j on a-C. We can always assume F is trivial over Vi' so nVi> ~(F»;;; nV j , ~). Thus we consider cPj as a holomorphic function on Vj - p. By Hartog's theorem cPj can be extended to all of U j ' Then P*cpj is defined on all of OJ and must equal (by continuity, or the identity theorem). Thus C{Jj(p) = O. Hence we have found cPj E r(V j , ~(F - p» such that'" j = P*C{Jj. We have proved that there is a o-cochain {cp;}, cPj E i' //) such that'" j = P*CPj, and thus P*CPij = P*cpj - P*cp j ' But P is surjective, so C{Jij = CPi - CPj· Thus {CPij}"" 0, and Hl(O/t, //) = O. Q.E.D.

"'j

nU

REMARK.

Relations between Hq(M, //) and Hq(M, IJ) are not easy to see.

To prove Proposition 8.1 it now suffices to prove HI(M, ~(p - C - D) = O. Let [C] and [D] be the corresponding bundles of the divisors C and D. Then we must show that Hl(M, ~(p - [C] - [D]» = O. To prove this it suffices to show that F - [C] - [D] - K(A:!) is positive, and then quote the vanishing theorem. We want to show that F - [C] - [D] - K(M) > 0

if m is sufficiently large, where F = mE, and K(M) is the canonical bundle of A:!. Therefore we would like to compute c([C]) and c([D]). First we find a l-cocycle on A:! representing [C]. Let z be a coordinate chart map centered atp E M, and let V = {zllzl < 2E}. Let P: A:! ~ M. Let us describe the normal bundle W of C in A:!. Let VA

where Ul' "',

Un

= {u E pn-11 U = (Ul' "', Un),

UA

#: O},

are homogeneous coordinates for pn-l. Then

and n

W=

U(VA X

C),

1=1

where we identify (u,

WA)

and (v, wI') if and only if U

= v,

(3)

We could define P: W - U by

(4)

8.

139

HODGE MANIfOLDS

Then pn-l = U). (V). x {O}) s;;; Wand we can identify C = Qp(p) with pn-l in W. Thus we consider a small neighborhood p-l(U) = of pn-l in Was a small neighborhood of C in M. On each VA x C, C is defined by IV). = O. Let OJ. = n (VJ. x C) for it = I, ''', n, and let 0 0 s M - C be such that

a

a

QpQq(M)

We set

Wo

=I

on

00 ,

=M = 0 0 u 0 1 u .. · u

Then the line bundle [C] is given by the I-cocycle on

Then (5) implies

gJ.O

an.

OJ. nOv.

(5)

= w)..

Recall that, in general, if F is defined by {Fjd and if aj Ifj kl 2 = ak for positive COO functions {at}, then i _ c(F) ,..", - iJ log a .. 2lt J

a

(6)

We want to find such COO functions for C. We make use of a Coo function ex on M with the properties (1) (2)

ex(z) = Izl2 for z E U, Izl < e ex(z) = 1 for z E M - V.

We define Ao(w) = cx(P(w», w E cx(z) A;.(w) =

Notice, on

Iw).1 2 '

WE

00

_ VJ..

en A). local coordinates are (it :F 0)

( ~ , ••• , U).-l , U).+ 1 " •• , Un) . U). U). U). U). Thus,

so the definition has meaning, and

A;.>OonU).

A. = 0, "', n.

140

GEOMETRY OF COMPLEX MANIFOLDS

The A;. satisfies so

i a0 log A;.. 2n

(7)

c([C]) '" -

We notice that

a0 log A;. = a0 log(1 + ...L.1. IU'U.I.1 2 ) and this is just the standard Kahler metric on pn-l = C. We also remark that

a0 (X(z) is a C 2-form on M of type (1, 1). Since a0 = d(o(X) (X

a 0 (X is cohomologous to zero on M, and the induced form

n = p*(ao(X) is

cohomologous to zero on M. We then define

Then 0" c '"

(8)

c( - [ C])

by (7), and in a neighborhood of C,

2niO"c

=

Uv 2 aa10g(1 + v";' L IU;. I)

+ ao(lzI 2 ).

Recall that Z

W;.

= (z l' ". ' n z) = -u = (. .. , wA, ... ) . U;.

Then

aO(L ZvZv) =

L dz v

dz v = dw;. " dw;. + .. '. "

Hence O"c is positive definite in a neighborhood of C. We get similar results for D. Next we want to find a relation between K(M) = K and K(M). We prove: PROPOSITION 8.2. K(M) = K + (n - 1) [C] + (n - 1)[D], where bundle over M induced from the canonical bundle K of M.

Proof

Suppose

M=

K is the

Qp(M). It is sufficient to prove

K(M) =

K + (n - 1) [C].

(9)

8.

HODGE MANIFOLDS

141

We choose U 3 P as in the previous proof. Then we choose {U j , Zj} coordinate systems on M such that {U} v {U j} covers M. Let (Zl, ... , z") be a coordinate system on M. Then the canonical bundle K of M is defined by the l-cocycle {J jk} where

dz) /\ ... /\ dzj =

J;;.' dzl/\ ... /\ dz~

dz l

J;'/ dzl/\ ... /\ dZ k

/\ ... /\

dz"

=

On M we may use {OA' OJ} as a coordinate covering where OJ = P-l(V) = Vj and 0 = p-l(U) = vi= lOA' using the notation of the previous theorem. On A we have the local coordinate system

a

and .. Z

WAU ..

=--,

if ex =I- A.

UA

ZA

= wA,

if ex = A..

Computing, we get

since

dz" =

d(W~~") = w

A

d(::) + (::) dw A •

Let {Ijk' [A., [Ak} be the Iacobians on M [which are used to define K(M)]. Then [ -I

jk

-1 [ Ak

=

J-I jk

1

WA

so

This proves (9) since

WA =

--1

= -;;-=-i J ok

0 defines c.

142

GEOMETRY OF COMPLEX MANIFOLDS

We now return to the proof of Proposition 8.1. Recall that E is positive, that is,

L '}',.p dz" "

c(E) '" '}' = - i

dz ll ,

where (y,.lI) is positive definite. For simplicity we write y > 0 or E> O. We want to show

p-

[C] - [D] - K(M) =

for large m. Let c(K) c(p -

K-

K.

F- K-

n[C] - n[D] > 0

(to)

Then

n[C] - n[D])

-my -

K + nUe

+ nUD,

where y = P*y and K = P*K. We choose m so large that my - K is positive definite on M. Then my - K is positive semidefinite on M and is positive definite on M - C - D. But Ue > 0 near C and UD > 0 near D. Then Equation (to) follows. This proves Proposition 8.1, and thus part (1) and (2). REMARK. It is an easy compactness argument to see that one can find an integer m such that HO(M, mE) separates points for all p, q E M, P "# q.

The proof of C is almost the same as the proof of B. We want to show that lI> is biholomorphic at each p EM. Consider f/' = (f)(F - 2p) which is the sheaf of germs of holomorphic sections of F which vanish at p up to order 2. Again we compute the stalks f/'z and write down the exact sequence 0-!/-Q(F)-!/"-0,

z"# P

!/z = (f)(F)z, !/p

= {cp I cP = CPj' CPj(z) = k,

L

ak • ••• kn z;' ... zJ", cP

E

(f)(F)p}.

+"'+k"~2

Then

!/; = 0, =

Thus HO(M, f/'II) ~

ifz"# p

{cp I cP = Qo +

c+ 1.

t a,. z,.},

,.=1

if z = p.

We write down the exact cohomology sequence

It is easily seen that to prove lI> is biholomorphic at p we need only show Hl(M, !/) = O. To prove this we once again use M = Qp(M), C = Qp(p).

8. LEMMA 8.2.

143

HODGE MANIFOLDS

If HICM, (!)(F - 2[C]» = 0,

then HI(M, (!)(F - 2p» = O. Proof The proof is the same as that of Lemma 8.1. One only has to notice that if (() has a zero of order 2 at p, P*({) has a zero of order 2 (at least) on C and vice versa.

LEMMA 8.3. Proof

HI(M, (!)(F - 2[C]» = 0 if m is large enough where F = mE.

Using Proposition 8.1. we find

F - 2[C] - K(M) = m£ =

K-

(n - l)[C] - 2[C]

mE - K - (n + 1)[C].

Hence c(F - 2[C] - K(M»,..,

if m is large enough.

my -

K

+ (n + l)uc >

0

Q.E.D.

REMARK. We again use compactness to see that there is an m which will work for all P E M. This completes the proof of Theorem 8.2. We now derive some consequences: THEOREM. 8.3. [Kodaira (1960)] If M is compact Kahler and H2(M, then M is projective algebraic. Proof

(!) =

0,

The exact cohomology sequence of 0-1L-(!)-(!)*-0

yields ···----.Hi(M, (!)*)~H2(M, 1L)-0.

Thus everything in H 2 (M,1L) is the Chern class of some bundle. Let {bl' ... , bm } be a basis for the free part of H2(M, 1L) so that H2(M, C) = Cb l

+ ... + Cbm •

144

GEOMETRY OF COMPLEX MANIFOLDS

Each b). = c(F).) and hence is cohomologous to a real 2-form of type (1, I). Let

w=

iL gll.lI dzll. "

dzll

be a Kahler form on M. We wish to modify w to get a Hodge metric on M. Since w E H 2(M, C)

W-LP).b). where PA E R (w is real and the b). are real). Given e, we can always find integers k)., r E 1L such that A. = 1, ... , m.

But then for a small enough e I

W = W -

~ - kJ.) Y). t... (PJ. -r

defines a Kahler form on M where yJ. - bJ. is a real (I, I) form. Hence w= rw' is also a Kahler form. But

Thus

w defines a Hodge metric on M, and M

is algebraic.

Q.E.D.

Theorem 8.4. [Kodaira (1954)] Let M be a compact complex manifold. If the universal covering manifold M is complex analytically homeomorphic to a bounded domain fJI £; Cn , then M is algebraic. Proof We make use of the Bergmann metric on fJI [see Helgason (1962)]. We have M = fJI/G where G, the set of covering transformations of fJI, is a collection of biholomorphic maps from fJI to fJI. Let ds 2 = L gll.lI dzll. dztJ be the Bergmann metric on (fl. We claim

(I) ds 2 is invariant under G and hence induces a metric L gll.lI dza. dz tJ on M = fJI/G. (2) If w = (i/2n) L ga.lI dza. " dztJ, w'" c( -K); so we have a Hodge metric on M.

This gives the theorem, thus we need only prove (I) and (2). Let .Yf be the Hilbert space of aU holomorphic functions f on fJI which have bounded norm

IIfII2 =

f If(zW dX, 91

8.

145

HODGE MANIFOLDS

where

dX = dX 1

•••

dX2ft and

= X2a:-l

Za:

+ iX2a:.

Let {I.} be any orthonormal base of JIf. Then the Bergmann kernel K(z, i) is given by co

K(z, i) =

_

L 1.(z)/.(z)

[= K(z)].

."'1

The kernel K(z) is actually independent of the choice of orthonormal basis {Iv} [see Helgason (1962)]. Then L ha:p dza: diP is a positive definite Hermitian metric where = 02 log K(z) hliP() z :l:l-. uZez uZp

Let 'I : f!J LEMMA

-+ f!J

be a biholomorphic map, y(z) = z'.

8.4. K(z)

= Idet a(z~, ... , z~) 12 K(z'). O(ZI' ••. , Zft)

Proof co

_

L 1.(z)/.(z)

K(z) =

."'1

and

f I.(z')j~(z') dX' f 1.(z)/iz) dX = Ov).· =

~

~

Let

O(Z'»)

F.(z) = I.(z') det ( o(z)

and notice

Thus,

and {F} gives a new base. Hence, co

K(z) = v~tv(z)F.(z)

=

I

o(z') \2

det o(z)

I

£1(z') 12

= det o(z)

"'iJ.(z')Jv(z')

,

K(z).

Q.E.D.

146

GEOMETRY OF COMPLEX MANIFOLDS

Since G is a group of biholomorphic maps this proves (I). Now let K be the canonical bundle of M. Let n : B -+ BIG = M. Let U j be an open set in M on which a local inverse of n is defined, and choose one J,lj = n- 1 to use as a coordinate chart for U j (J,lj(p) E C" if P E Uj). Suppose P E U j n Uk. Then there is "Ijk E G such that J,lip) = "Ijk(pip». The canonical bundle K on M is defined by the l-cocyc1e

and we have K(Zj)

= l./jkl 2

K(Zk)

by the lemma. Recall that if we have positive Coo functions a j on U j such that

aj 1./j1c1 2 = ak , then i ~ "I c( - K) = - 2n 0 u log a j .

Therefore, let aj

= K- 1(zJ Then

c( -K) =

~ oJ log K(zj) = ~ L glJll dzj " dz~. 2n

2n

This proves the theorem. There is much interest in nonalgebraic Kahler manifolds. Kahler manifolds give examples of the minimal surfaces of differential geometry.

REMARK.

[4] Applications of Elliptic Partial Differential Equations to Deformations I.

Infinitesimal Deformations

We want to study analytic families of compact, complex manifolds. Informally, we are only interested in small deformations. We may as well assume our base space B, = {til tl < r, tEem} is an open disk around the origin of em. We want a manifold"'" and a holomorphic map w: "'" -+ B, with maximal rank so that w is proper and each fibre Mr = w-I(t) has the structure of a complex manifold which varies analytically with t. We want a covering {au j} of "'" so that

au j

e= j

t)1"jl < 1, It I < r} (e),···, ej), w(e j , t) = t,

= {(e j

,

and

Cj = fMe j , t) on au j n auk'

e

wherefjk is holomorphic in j and t. We notice that under these circumstances Mr is diffeomorphic to M 0' and in fact, "'" is diffeomorphic to X x B, , where X is the underlying differentiable manifold of Mo. Thus au j = U j x B, where U j = {e j IC j < I}, and

M =

UU

j

x B,.

j

If x is a point of X, t E B, we notice that

ej = ej(x, t) is a differentiable function of (x, t) and we have

ej(x, t) = fjk(ek(x, t), t).

(1)

Let M = M 0 = X and use the complex coordinates z of M as differentiable coordinates so that

ej(x, t) = ej(z, t), where ej(z, t) is a differentiable function of z and t. Because t = 0, ej(z,O) is holomorphic in z (otherwise it is only differentiable). 147

148 DEFINITION

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

1.1.

Let

(a) (-ata) =Lc .=1· at. m

belong to the tangent space to B, at the origin. We define (aM,/at),=o to be the cohomology class in Hl(M, 0) given by the I-cocycle

(J;k=I afMek,t) .. = 1 at

I (';) . ,=0

aei

We want to represent {(J;k} with a I-form by using Dolbeault's theorem. Let T be the holomorphic tangent bundle on M and 0 = @(T) be the sheaf of sections. Then we have the resolution D

D

0 _ 0 _ A o ( T ) _ A o . 1(T)_···, where AO·q(T) is the sheaf of Cal vector (0, q)-forms. Locally, such a thing has the representation

where 1 rn P = .." q!

~ rn~

_

L... .." .. , .....q

dz-'" /\ ... /\ dz ..q

and

Let us trace through the Dolbeault isomorphism 1 r(oAO(T» H (M, 0) ~ or(AO(T» .

Let (aM,/at),=o -11. Then 11 is defined as follows: Pick such that Then

PROPOSITION

1.1. 11 = -

I

«= 1

o(amz, t) at

I )(';). ael ,=0

ei

E

r(U;, AO(T»

1.

Proof

INFINITESIMAL DEFORMATIONS

149

Let

Then Equation (I) yields

Ci = L Of~k er + (a fik )

,

at

a~

1=0

Thus,

If we set

ek = - L cf(ajoef),

- L oCi(ojaei).

we get

Bil:

= el: -

ei'

Q.E.D.

Therefore,

,,= oel

=

We want to define a vector (0, I)-form q>(t), t E B which describes the complex structure of M,. With respect to the local complex coordinate z on a neighborhood W of M we have the differential operators

a, (a~')' 0, (a:')' =

=

and

Recall that where fjt is a holomorphic function of el:' Thus

oej(z, t) = and

t ~~k oef(z,

t),

150

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

At t = 0, (Zl, .. " z") are local complex coordinates on M and «(J(z, 0), .. " (j(z, 0» are also local coordinates. Hence

So, for small enough

Itl,

Let

(

!l~«)-I u~·

A

Aj«= azi

.

Consider the local I-form,

L A1« a(j(z, t) =

qJ~(z, t).

«

We claim the form qJ~(a, t) is well defined independent of (j. For fjk«(k' t) holomorphic in (k implies

which yields A( z, t ) = ~ A oYII( L. ()YII Aj« .. ,. Z, t ) «.11 ':ok

oq

qJj

=

L Atll oCf(z, t). II

We therefore define cpJ. = cp~(z, t). Then qJJ.(z, t) is a (0, I)-form independent of (j' but it still depends on the local coordinate z. Let z = Zw. Suppose V is another open set in M with local coordinate z.,. Let qJ~ = qJJ.(zw, t) and qJ~ = qJl(Z." t). If we set

then cp(t) =

=

~ qJ~(zw, t)(a:~)

~ cpecz." t)(a:~)

l.

INFINITESIMAL DEFORMATIONS

151

is a well-defined, global vector (0, I )-form on M. Now by the definition of A?2'

~r«( t) = "~ cp ). az o'jA.

u'>j Z,

Therefore (2)

1.2. The complex structure on M, is determined by cp(t). More specifically a differentiable function f defined on any open subset of M is holomorphic with respect to the complex structure of M, if and only if PROPOSITION

(0 Proof

L cp'\t)op)f(z) = O.

(3)

We represent cpp(t) by

I

cpP(t) =

cp(t)fdz~

tl

If f satisfies (2), then

( 0« -

t cp( t)~ a

P)

f

=

for all cx.

0

(4)

We use

ol :l_P =

v.

I

Y

of o~}

of o'}

o~rY a-(I + Ll v'>j :lj'Y azP '>j"

and Equation (4) to get

Equation (2) implies that the first term is zero so

(5) Since 'j(z, 0) is holomorphic in z, cp(O) = small for small t. Thus,

o.

Continuity tells us that cpU) is

t

(Va - CP(t)~D/l)'~ is invertible for small t. Hence Equation (5) yields

a! = o.

o'j

152

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Thus f is holomorphic in Cj(z, t). We can read the argument backwards, so the proof is concluded. Q.E.D. We want to introduce a bracket operation in the algebra of vector 0, q)-forms. Let us recall the Lie bracket of vector fields. If u = ulJ OIJ , W = L ~OIJ' then [u, wJ = L(UIJO.. wp - wIJOIJUP)op .

L

... 11

For the generalization, let

DEFINITION

1.2.

"

[cp,I/I]= L (cpIJ A a.. I/I II -(-I)pQl/lIJ Ao.. cpll)oll IJ. fJ= 1 where a.. .1.fJ 'I' -

PROPOSITION

1

_

"

;:0

p.,~u..

cpfJ

_

AI''')'

dz-.l. 1

p

A

"

•••

A

"

dz-.l. p •

1.3.

I/IJ is bilinear [1/1, CPJ = -( -l)pq[CP, I/IJ (J[cp, I/IJ = [acp, I/IJ + (-I)P[cp, al/lJ (4) (-I)PTcp, [1/1, rJ] + (-I)QP[I/I, [r, cp]] + (_l}rq[r, [cp, I/I]J

(I) (2) (3)

[cp,

=

0,

where cp is a (0, p)-form, 1/1 is (0, q) and, r is (0, r). (This is the Jacobi identity.)

Proof Uninteresting; we leave it to the reader. We collect our facts into the following theorem: THEOREM 1.1. If iii : vH -+ Dr is a complex analytic family of compact complex manifolds, then the complex structure on M, = iii-let) is represented by a vector (0, I)-form cp(t) on M o such that

=

°

(1) (2)

ocp(t) - t[cp(t), cp(t)] cp(O) =

(3)

(OM,) __ '1 = _ (oCP(t»)

°

ot

,=0

ot

E 1=0

quAo(T».

1. Proof

INFINITESIMAL DEFORMATIONS

153

(2) has already been done. As for (1), by Equation (2)

(~cpP(t)iJp'j -

= o.

O,j)

So

o = o(~ cpP(t)Op,j) =

L ocpP(t)opej - L cpp(t) A oOp'j. P

P

Thus,

L OcpP(t)Op 'j = L cpp(t) A Op o'j P

P

~ cpfl(t) A OP(~ cp(t)YOy Cj) = L L cpfl(t) A OflCP(t)Y· Oyej

=

P y

+ L L cpp(t) A

cp(tYOpOy'j.

P y

The last term in this expression is zero, since cpP(t) A cpY(t) is skew-symmetric in p, y and op Oy Cj is symmetric in p, y. So we have (6) Now det(op ej(z, t» is nonzero if t is small, so afl ej is invertible for small t. Then (6) yields

ocpp(t) =

L cpY(t) A OycpP(t) Y

= ![cp(l), cp(t)JI1,

since

[cp, cp]p = L cplZ A OIZCPP - (_l)lcplZ A OlZcpP = 2

L cplZ A OlZcpl1.

Finally, for (3), we already know '1 = -

~ O~~(a~~)

where

,~ = (oe~) at

. 1=0

154

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

But

and the last term is zero, so

= _

.

(a~(t)) of

Q.E.D.

1:0

We wish to find conditions on a cohomology class p for it to represent an infinitesimal deformation. We first check to see that the bracket [,] extends to cohomology. Let

cp E qaAO,p-l(T»,

t/J E qDAo.q-l(T», so that

Then it is obvious

acp = at/J = O. that D[cp, t/J] = O. Also if t/J = D(1, then [cp, t/J] = [cp, C(1] = ±o[cp, (1].

Hence, Eo] induces a map HP(M, 0)<8> Hq(M, 0)~ Hp+q(M, 0)

by using Dolbeault, and the facts just noticed, qDAo. p-I(T» HP(M, 0) ~ ur(Ao. P-I(T»'

Now let (tl' .. " fm) be coordinates on Br . Then any infinitesimal deformation is a linear combination of ones of the form

We claim: THEOREM 1.2.

If P E HI(M. 0) is an infinitesimal deformation, then

[p,p] =0.

2. Proof

AN EXISTENCE THEOREM FOR DEfORMATIONS I

155

By using the previous remark we need only check that ['1,., '1)] = 0

for all A, v. We differentiate the equation

ccp(t) - [cp(t), cp(t)]

=0

twice at t = 0 to get

= 2

acp aCP] [at;.' at•.

Let

Then we get

Q.E.D. We remark that this is not a sufficient condition, and the higher derivatives give more information. It is quite difficult to compute [p, p] for a general p E Hl(M, 0), but in most specific examples we get [p, p] = O.

2. An Existence Theorem for Deformations I. (No Obstructions) We aim to prove the following theorem: THEOREM 2.1. [Kodaira, Nirenberg, and Spencer (1958)]. Let M be a compact complex manifold. Assume that H2(M, 0) = o. Then there exists a complex analytic family vH ~ B., where

B. = {tlltl < e} ~

em, In =

dim Ht(M, 0),

such that: Mo = w-t(O) = M. (2) The map To(Bt) -+ Ht(M, 0) given by (D/Dt) -+ (iJM,/iJt),=o is surjective (in fact, an isomorphism). More specifically if {Pl' ... , Pm} is a base

(I)

156

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

for Hl(M, 0), then we can define ..A so that

(OMI) ot. Proof

(1)

=

P

1=0

for v = 1, "', m.

•

We will accomplish the proof in the following two steps:

Construction of a vector (0, I)-form

m(t) = "L.. 'f'k""km m tk' 'f' 1

••• t km In

such that

cp(o) = 0, ocp(t) - t[cp(t), cp(t)] = 0, and = P. E Hl(M, 0). ( O~(t)) ot. 1=0

(2) Show that cp(t) determines a complex analytic family by using the Newlander-Nirenberg theorem. First we survey the Newlander-Nirenberg theorem, which is sometimes called a "complex" Frobenius theorem. Let U !;;;;; en be an open domain, and

cp =

L cp~ dZ"(o~1I )

a vector (0, I)-form on U. Let

La =

(o~~) -IItl CP~(Z)(a~1I ).

We want to consider solutions to the equations Laf(z)

=

°

(1)

on the domain U. The theorem [of Newlander and Nirenberg (1957)] is: THEOREM.

If Li and Ii are (complex) linearly independent, and if

ocp - t[cp, cp] = 0, then Equation (I) has n COO solutionsft(z), .. ',J,,(z) such that det(O(fl' .. "

O(Zl> "',

In' I}, .. " In)) ~

°

Zn' Zl' " ' , Zn)

(that is,ft, ... ,J" define a differentiable coordinate system on U).

2.

AN EXISTENCE THEOREM FOR DEFORMATIONS I

La will be linearly independent.

REMARK

1.

If t is small, cp(t) is small and L i

REMARK

2.

Linear independence is needed; for if

,

157

iJ 0 0 L-=-=-+cz

az"

oxcz

OZ"

then La! = 0 implies! is independent of x". REMARK 3. If M is a complex manifold and qJ is given satisfying the conditions of the theorem, then by using (the proof of) Proposition 1.2 we see M has another structure as a complex manifold which is described by the form qJ. We say the almost complex structure qJ is integrable, and hence associated to a complex structure.

In order to construct our form qJ(t) we need to do some more potential theory. We want to define the Green's operator on

,f£q = r(Ao·q(T))

= the space of vector (0, q)-forms.

To do this we introduce an Hermitian metric g«ll on M, and define an inner product,

where the * operator has been defined before. We have the adjoint B of 0, (BqJ, 1/1) = (qJ, 01/1); and the Laplacian 0 = Bo + oB. Then the space of harmonic forms

IHI q

= {qJ I qJ E ,f£q, OqJ = O} ~

Hq(M, 0),

defines a Hodge decomposition,

,f£q

= IHI q liB O,f£q::2 W liBo,f£q-1 + B,f£q+ 1

into an orthogonal direct sum of subspaces. Thus, for qJ E ,f£q, qJ = " " E

IHI q,

1/1 E ,f£q.

+ 01/1,

Since 1/1 E ,f£q,

1/1 = , + 1/11' ,

E

IHI q,

1/11 E O,f£q,

and

Thus (2)

158

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

LEMMA

2.1.

Proof

The decomposition in Equation (2) is unique. Surely '1 is unique. If 1/1',1/1 are both orthogonal to IHI q and

ep = '1 + 01/1', ep = '1 + 01/1, then

0(1/1' - 1/1) = 0 and

I/I'-I/IEW. But

1/1' - 1/1

.L

IHI q so 1/1' = 1/1.

Q.E.D.

DEFINITION 2.1. Given ep, the unique 1/11 making Equation (2) true is denoted Gep, and the mapping ep -+ Glp defines G : ,Pq -+ o,Pq. G is called the Green's operator, and is a linear map. We write '1 = Hep and call H the harmonic projection operator. Then

+ OGep.

ep = Hep PROPOSITION

2.1.

oH

=

Ho

= 0,

9H

=

H9

(3)

GH

= 0,

=

HG

= 0,

oG

=

Go,

9G = G9.

Proof

oHep = 0 since Hep E W

=

{I/I I 01/1 =

91/1 = OJ.

Hoep = 0 since oep E o,Pq .L W. The proof that 0 HGep = 0 since Gep .L W. For GH = 0 notice Hep = HHep

=

Hep

=

9H

=

H9 is analogous.

+ OGH IP

and uniqueness yields GHep = O. The proofs of the last two are similar to each other so we only prove the first of them. Recall 00 = 0(09

+ 90) = 090

and

Do = 090. Thus 00

= Do and oep = 00 Gep = OoGep = Hoep = o Goep.

2.

AN EXISTENCE THEOREM FOR DEFORMATIONS I

159

Since oGcp ..L Wand Hocp = 0, we use uniqueness of decomposition Equation (3) to see oGcp = Gocp. Q.E.D. To proceed further. we need to introduce the Holder norms in the spaces fi7 q • To do this we fix a finite covering {Vj} of M such that (Zj) are coordinates on U j • Let cp E fi7 q ,

~ CP~(Z)(a:~)

cp =

1

cp j). -- q! "L.. cpAjii,..

ii.

dza, j

1\ .•. 1\

dz"' j .

Let k E 7L, k ~ O;!X E~, 0 < !X < 1. Let h = (hi' "', h 2n ), hi ~ 0, 2}~1 hi where n = dim M. Then denote

z} = Then the Holder norm

Ilcpllk+a. =

X;a.-l

= Ihl

+ ixj.

Ilcpllk+a. is defined as follows:

m~x{ Lh (sup ID~cpfci, .. ii.(z)l) :eV) J

Ihl ~k

+ J1,,,,Vj sup

ID~ CP;a, ... iiq(Y) - DJ cpfa, ... !i.(Z)I)

Iy-zl

Ihl =k

where the sup is over all A, !XI' ••• , of DougHs and Nirenberg (I 955).

!X q •

a . '

(4)

We have the following a priori estimate

(5) where k

~

2, C is a constant which is independent of cp and

IIcpllo =

max A,

sup ze Vj

j (lIt"

I

Icp;il, ".il (z)l. •

12q

REMARK. One can see that two norms defined as in Equation (4) for two different coverings {U j }, {Uj} induce equivalent topologies on gq.

PROPOSITION 2.2. dent of cp and "'.

Proof

II [cp, "'] 1Ik+. :::;; C Ilcpllk+ 1 +II IIcpllk+ 1 +/%' where C is indepen-

We leave the simple check to the reader.

We need to know the following strong kind of continuity for the Green's operator G:

160

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

2.3. not on
PROPOSITION

k and

IX,

Proof

IIG
~

2, where C depends only on

We use Equation (5) to get

IIG
+ IIG
c(11

H
The space IHI q is finite dimensional, so let {b.}:=l be a base. Then m

H

'L (
.=

1

so m

11
'L

.=

Ilfl.llk-2+ .. max I(
1

•

~

C1('L IIP.llk-2+ ..)II
~

C2 11
Thus,

(6) and we need only prove

that is,

IIG
(7)

Suppose (7) is not true. Then there is a sequence
.

hm v

IiG
By mUltiplying
1I
(is bounded for k Write

~

2).

(8)

2.

AN EXISTENCE THEOREM FOR DEFORMATIONS I

161

Then Equation (8) implies that each of (G(p<·»)j«.".«.

and all of its partial derivatives up to order k are uniformly bounded and equicontinuous. We are in a position to use Ascoli's theorem. We can choose a subsequence {cp('n) In = I, 2, ... } such that Gcp(v n) and Gcp('n) converge uniformly to t/I and D~ l{! for Ihl ~ k. For simplicity let us replace Vn by v. Then we get

D'

(t/I, t/I) = lim(Gcp('), t/I)

= Iim(Gcp('), DGt/I + Ht/I) (since Gcp(') .lW) = lim (Gcp(V), DGt/I) v"'''''

(self adjointness) = lim(DGcp('), Gt/I) v"'''''

= Iim(cp(') - Hcp('), Gt/I)

.... ""

(since cp(,) ~ 0)

= O.

Thus t/I = O. But we should have "t/llio = I, since "t/lilo = lim "Gcp(')ilo = 1.

.... ""

This contradiction proves the proposition. Let us now begin to construct the cp(t) of Part (a). We use power series techniques but we notice that we could also use the implicit function theorem for Banach spaces [compare Kuranishi (1965)]. We want to construct cp(t) = 1 cp,lt), where

L:=

L

cp,.(t) = "I

CPVl ". Vno t~' ... t~m

+ ... + "m:OJl

and each CP" ". "ft, E r(Ao,I(T)) such that

ocp(t) - t[cp(t), cp(t)]

= 0,

(9)

m

cp!(t) =

L '1. t., ,.=

(10)

!

where {'1.} is a base for IHI! ~ Hl(M, 0). We use a method due to Kuranishi. Consider the equation

cp(t) = cp!(t)

+ -l8G[cp(t), cp(t)],

(II)

where cp!(t) is given by (10). We first show that (I I) has a unique formal

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

162

power series solution cp(t). In fact, this is clear since CP2(t)

= !.9G[CPl(t), CPl(t)]

CP3(t) = !.9G([CPl(t), CP2(t)]

+ [cpit), CPl(t)])

cp,,(t) = t.9GCt: [cp).(t), cP,,_;.(t)]).

PROPOSITION 2.4.

II

For small Itl, cp(t) =

(12)

L;:'= 1 cp,,(t) converges in the norm

1Ik+«' Proof

Let

- "~ A VI

-

"Vm

t VI 1

•••

Vm t III •

As usual. Ilcp(t)llkH ~ A(t) means IIcp'I" v", Ilk+«::; A' I .. v",, and cp"(t) = CPl (t) + ... + cp,,(t). Then (I2) is equivalent to (13)

We want to choose fJ and y. Suppose they are chosen so that IIcp,,-I(t)lIkH A(t). Since .9 is a linear differential operator of first order,

~

II1.9G[cp, I/J]ilk+a ::; C I ilG[cp, I/J]IIH 1+« ::; C1Ck,all[cp, I/J]lik-l+a ~ C't Ck , II C IIcpllk+« IlfiJllk+a'

by Propositions 2.2 and 2.3, where Cl ' Ck • a' and C are constants independent of cp and I/J. Hence by (13)

I cp(/ll( 1) Ii k +" ::; C I Ck ... C 11 cp(" - I l( t)IIH« II cp(/l - I l( t)11h+ .. ~ C 1Ck , .. C(A(1»2

~ CICk,,,C(~)A(t) as in Section 3, Chapter 2. Thus choose fJ and y so that C1 Ck,a. c(fJ/y) < I. Then Ilcp"(t)IIk+a. ~ A(t). These constants are all independent of JJ.. So if fJ and yare chosen so that Ilcpl(t)llkH ~ A(t), which is clearly possible, then C1 Ck ... c(fJ/y) < I yields

I cp(t)llk+a ~ So for small Ill, cp(t) converges.

A(t).

Q.E.D.

2.

AN EXISTENCE THEOREM FOR DEFORMATIONS

I

163

PROPOSITION 2.5. The q>(t) of Proposition 2.4 satisfies oq>(t) - Hq>(t), q>(t)] = 0 if and only if H[q>(t), q>(t)] = 0, where H: reAo. 2 (T» -+ 1HJ2 ~ H2(M, 0) is the orthogonal projection to the harmonic subspace of 22 = reAo. 2 (T».

Proof If oq> = Hq>, q>], then 0 = Hoq> = !H[q>, q>], since Ho = O. Conversely let H[q>, q>] = 0 and set I/I(t) = oq>(t) - t[q>(t), q>(t)]. Then each 17. E W so 017. = 0 and

21/1(t) = o.9G[q>(t), q>(t)] - [q>(t), q>(t)]. Recall that any w can be decomposed w = Hw + DGw. Since H[cp, cp] we get 21/1(t) = (u.9G - DG)[cp(t), q>(t)]. Because D =0.9 +.90 we get

= 0,

21/1(t) = - .9oG[cp(t), q>(t)]

=-

.9Go[cp(t), cp(t)]

= -

29G[oq>(t), q>(t)].

This last equality is true because

o[q>, cp] = [oq>, q>] - [cp, oq>] = [oq>, q>]

+ [oq>, q>]

= 2[ocp, cp].

Then

cp(t) = -.9G[oq>(t), cp(t)] = -.9G[I/I(t)

+ ![q>(t), q>(t)], q>(t)]

= - .9G[I/I(t), q>(t)] by the Jacobi identity. Estimating, we get

i 1jJ( t) il k +

C, C k.
Choose It I so small that IIq>(t)IIkH C, Ck.
q>(z, t) is Coo in (z, t) and holomorphic in t.

Proof It is immediate that cp is Ck since the series converges in II IIk+
164

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Since the 11y are harmonic,

Oep

= to.9G[ep(t), ep{t)] = t.90G[ep{t), ep{t)] =

-!.9H[ep{t), ep(t)]

+ t.9[ep(t), ep{t)]

= t.9[ ep(t), ep(t)].

Since ep is holomorphic in t,

Thus ep satisfies (14)

Equation (14) is quasi-linear elliptic since m a2 L at- +0 at

y=

1

y

y

is elliptic and .9 is first order. We know that cp(z, t) is small for small t and that the coefficients of (14) are COO. Under these circumstances the regularity theorem says that solutions of (14) such as cp(z, t) are Coo. Q.E.D. So with the assumption that H2(M, 0) = 0, the Newlander-Nirenberg theorem implies that each ep(t) defines a complex structure M, on M. Thus we obtain a family {M,l It I < e} of complex manifolds.

PROPOSITION

2.7.

{M,l

It I < e}

is a complex analytic family.

L

Consider ep(t) = epg(z, t) dz«{a/az P) as a vector (0, I)-form defined on M x Be where Be = {tlltl < e}. Then ep(t) satisfies the integrability condition 0 = oep(t) - t[ep(t), ep(t)] on M x Be where Proof

oep(t) =

f y=

acp~) drY + 1

01

i

aep_~) dirt

rt= 1 az

and aep{t)/al· = 0 since cp is holomorphic in t. Thus ep determines a complex structure vi{ on M x Be. The local complex coordinates of .Ii are solutions , of (15)

3.

AN EXISTENCE THEOREM FOR DEFORMATIONS II

165

Equation (15) is satisfied if and only if

a

a(

-(-~,>pP--o vz" fJ ii az P ,

o:=t,"',n (16)

ac

ai v =

0,

v= 1, ... , m.

Hence, on some coordinate chart o/i j = U j X BE £; .A we have n + m independent solutions t), ... , t m , (j(z, t), ... , ej(z, t) of Equation (15). So ""{ is a complex manifold such that the projection 11: : .A -+ BE is holomorphic of rank m and for each fixed t, 1I:- 1 (t) = M, is the complex manifold with complex structure given by M,. Q.E.D. This also completes the proof of Theorem 2.1. We remark that by the completeness theorem (see Chapter 2, Section 3) the family 11: : .A -+ BE is complete at 0 E BE'

3. An Existence Theorem for Deformations II. (Kuranishi's Theorem) We want to discuss the case in which H2(M, 0) does not necessarily vanish. Again fix an Hermitian metric on M and define 8, 0, G, "', and so forth. Let {'lv I v = I, "', m} be a base for W ~ Hl(M, 0). It is not necessary to know HZ(M, 0) = 0 for Proposition 2.4 to hold. Thus we still have a unique convergent (even in the norms defined later in this section) power series solution fP(t) of fP(t)

= "lCt) + t8G[fP(t), fP(t)],

(I)

m

where 'let)

=L

'lv tv' And this fP(t) satisfies

v= 1

OfP(t) - ![fP(t), fP(t)]

=0

(2)

if and only if H[fP(t). fP(t)] = O. Let {{3 AI A. = 1, ... , r} be an orthonormal base of \Hl 2 and let (,) be the inner product in A2 = r(AO. 2 (T» (note the change of notation from Section 2, A 2 +-+ 2 2 ). Then r

H[fP(t), fP(t)] =

L ([fP(t), fP(t)], {3A){3A'

(3)

A=1

Hence H([fP. fP]) = 0 if and only if ([fP(t), fP(t)], {3).) = 0 for A. = I, "', r. Since fP(t) is a power series in t so is ([fP(t), fP(t)], {3).) = b).(t). Thus b).(t) is holomorphic in t for A. = I, "', rand It I small (It I < e). Also b).(O) = O. Define an analytic set S as follows:

S = {tlltl < e, b).(t) = 0, A. = 1, "', r}.

166

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Then S is an analytic subset of B. containing the origin and we have proved the following proposition:

cp(t) satisfies ocp(t) -

PROPOSITION 3.1. COROLLARY.

{M,

It E S}

t [cp(t),

cp(t)] =

°

if and only if t E S.

is a set of complex structures on M.

We note that S may be singular; however, {M, It E S} can be made into a complex analytic family over S. The details are in Kuranishi (1965). We want to prove that if 1JI is a small enough vector (0, I )-form such that 01/1- 1[1/1, 1/1] = 0, then M", is biholomorphically equivalent to one of the M" t E S. This is the completeness theorem (an abbreviated version) of Kuranishi. We find it convenient to use the Sobolev norms since some of the continuity properties that will be needed will be more transparent in these norms. For an open set U of IRft and complex Ceo functions f and 9 defined on U, the closure of U, we set

L

l"lsk

f Dj(x) . D"g(x) dx,

(4)

U

where we use the multi-index notation (X = «(Xl' "', (Xn), (Xi ~ 0, I(XI = and D" = (a/ax i ),,' ••• (atax")"". Then

II/ilk = J <1,J)k = Ilfllf.

I?=

i

(Xi' (5)

The classical Sobelev lemma states that if V is a relatively compact open subset of U, then there is a constant c such that if x E V and k > n/2

ID"/(x)1 ~ c Ilfll~+ 1"1'

(6)

where c is a constant depending only on k, I(XI, V, (6) we have: LEMMA 3.1.

There is a constant c such that if k

Ilfglli Proof

Let

It I ~ k.

~

~

u.

n

As a consequence of

+ 2,

c Ilfll~ . IIglif.

(7)

Then D(jg)

=

L

D'j' D'g.

r+.=(

Then either Itl ~ k nd k that

Irl + nil + I ~ k or lsi + nil + I ~ k when r + s = t, since + 1 implies It I + n + 1 ~ 2k. Thus, there is a constant K so

~ n

IDflg(x)1 ~

L

K r+.=f

(lIfIIf ID'g(x)1 + ID'j(x)llIglI~)

3. for x

E

AN EXISTENCE THEOREM FOR DEFORMATIONS

167

II

V, by Equation (6). Squaring,

IDtfg(xW

~ ,,' L+~=}llfIIf)2IDsg(xW + (1Igllf)2IDrf(XW].

Now (7) follows easily.

Q.E.D.

By using a partition of unity, one defines IIcpllk for any cp E A P = reAD. P(T». The estimates of the previous section are essentially the same and we list the ones we need.

II[cp,

t/I] Ilk ~ ckllcpllk+ Ilit/lllk+ 1 [for alllarge k. (k

~

2n

+ 2where dime M = n).] (8)

IIHcpllk ~ Ck Ilcpllk

(9)

lI.9gcpllk ~ ck Ilcpllk-l·

(to)

From now on k will be chosen so that the conditions of Lemma 3 hold.

PROPOSITION solution. Proof

3.2. Let

T

For fixed '1(t), Equation (I) has only one small

(1lcpllk < e)

= cp - cp(t). Then

• = t.9G([cp, cp] - [cp(t), cp(t)]) = t.9G([., cp(t)] = t.9G(2[.,

Estimating

+ [cp(t),.] + [T, T])

cp(t)] + [" .]).

11.llk gives 11.llk ~ D(II'llk Ilcp(t)lIk + 1I'lIi) ~ D IITMllcp(t)llk + 1I.llk).

If

IIcp(t)IIJ. is

small enough, the only way

o~ x ~

Dx(lIcp(t)lIk

can happen with x close to 0 is for x =

L

o.

+ x) Q.E.D.

The set N = {'1(t) = '1v tv Iitl < e} describes a small neighborhood of For small enough B there is a one-to-one correspondence between '1 E N and solutions cp of (1). This is proved as follows: If cp = '1(t) + -t.9G[cp, cp], then H.9 = 0 implies '1 = Hcp. Thus '1 is uniquely determined by cpo Given '1. call the small solution of Equation (1) (given in Proposition 3.2) cpo Then a correspondence F is defined by F'1 = cpo

o E IHJI •

168

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

LEMMA 3.2. Suppose M", is given, If 8", = 0, then'" = qJ(t) for some t if 1I"'lIk < tJ for small tJ.

Proof

0'" - t["" "'J = 0, hence

0'" = =

80'" + 08", t8[""

"'J

since 8", = O. Then

'" - H", = GO'" = tG.9[""

"'J.

Let 11 = H "'. Then '" = 11 + t8G[ "', '" J; and by assumption II'" I k is small so that 1II111k is small by Equation (9). Hence 11 = 11(t) for some It I < e, and the Q.E.D. remark .iust made shows that'" = FI1 = Fy/(t) = cp(t).

In general 8", =f: 0 so we must try something else. When we say that M '" is holomorphically equivalent to M ",(t) we mean that there is a biholomorphic mapf: M",(t) ~ M", andf: M ~ M thus induces a diffeomorphismJ: M ~ M of the underlying differentiable manifold M. Conversely let f be any diffeomorphism such that its values and first derivatives are close to the values and first derivatives of the identity map. Cover M", with a system {Vj , 'j(z)} of local ",-holomorphic coordinates and let {Vj}, Vj £; Vj be a covering of M so thatf(Uj) £; Uj • Then 'j(f(z» is a local holomorphic coordinate on Uj. Thus qJ is determined by the equation n

o'IJ(f(z» =

L qJP(z)op,IJ(f(z».

(11 )

P=l

We also know n

o'lJ(z) =

L ",P(z)op,lJ(z), P=I

that is,

~ "IJ( ) = ~ .I,P( ) o'lJ(z) !l-A ~ z L... '1'1 Z _p. uZ

P=I

OZ

(12)

3.

AN EXISTENCE THEOREM FOR DEFORMATIONS II

169

Putting Equations (II) and (12) together we get

" aC"(f(z» ofP( ) + " aC"(f(z» "J). 1" af P z ~ ar 0 - " Y() z y.P

- l..J qJ

ac"fJ aY fP + "l..J qJY() ac" aY J). , af z aJ)' Y.).

The matrix (aC"jap) is invertible since it is assumed that Thus,

11"'"k is close to zero.

ofP(z) + L 1/11(f(z» op'(z) ).

PROPOSITION 3.3. Let M", be the complex structure induced from M", by the map f: M ~ M. Then qJ is determined by Equation (13). We use the notation 1/1 0 fto denote cpo

Proof

We just notice that

11"'llk small implies that

(ayfP(z) + ~ "'1(f(z»oyr(Z») is an invertible matrix.

Q.E.D.

Thus to prove Kuranishi's theorem in the case 9", =F 0 it suffices to show the existence of a diffeomorphismfsuch that 9(",0 f) = O. This is our task. We need to digress for a moment to describe a way of indexing diffeomorphisms close to the identity by the use of geodesics. We shall be brief, and refer the reader to any text on differential geometry for missing details. Let an Hermitian metric (g"jJ) be fixed on M. Then we have the Christoffel symbols

r"AfJ =" gji,,(OgfJji) l..J J. ' ,.

:l uZ

170

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

and if u(z)

=

L ull(z)(O/OZIl) is a vector field, we have the covariant derivatives t7

V;.U

-

II

OUIl '" rll u II =-;-;:+ i..J ;'lI , uZ

II

V;.u =

II

OUIl

oz;"

If z(t) is a curve in M, we define

d

dz;'(t)

= d ulI(z(t» + L r~lI(z(t» -d- ull(z(t» , t

t

;., II

when u(z) is defined along z(t). Then the geodesics of M are the curves z(t) satisfying the differential equation

V, (

dZIl(t)) = 0 dt '

that is, d(ZIl(t» -d 2

t

dz;'

+ ;.,LII r~lI(z(t»-dt

dz ll (t)-d (t)

t

(14)

= O.

Since M is compact, all metrics are complete, that is, the geodesics through a given point with given tangent direction are defined for all time t. Let Zll(t) = Zll(t, ZO, ~) be the solution of Equation (14) satisfying ZIl(O) = (dzll/dt)(O) = ~II. Then the following facts are easily verified by using existence and uniqueness theorems of ordinary differential equations:

zo,

(I) (2)

Zll(t, zo, e) are Coo in (t, zo, e). zll(kt, zo, e) = Zll(t, zo, ke).

We set JIZ(ZO' () = zll(l, zo, e)· Then J is COO in (zo, e) and J«(zo, te) = Zll(t, Zo, e). Differentiating this relation we get dz ll

dt

n ofa (t, Zo, e) = 1I~1 ell o~1I (zo, Ie)

n

_

Ofll

+ 1I~1 ell a~1I (Zo, te).

So

~II =

] dz n [ or ar dt (0, Zo, e) = 111;1 ell o~1I (Zo, 0) + ~II a~1I (Zo, 0) . ll

This implies

ar

or

r(zo, 0) = z~, oe ll (zo, 0) = Dp, a~1I (Zo, 0)

=

o.

3.

AN EXISTENCE THEOREM FOR DEFORMATIONS II

171

The Taylor expanison then yields (15)

where O(l~12) is a term bounded by M 1~12 for some M> 0 and for small lei. Given a vector field ~ = 2::=1 ~Il(z)(oloZIl) on M we define a diffeomorphism !c. : M --+ M by Zll --+ r(z,

~(z».

By Equation (IS) f~(z)

(16)

= Zll + ~1l(Z) + O(I~(zW).

( 16)

We wish to calculate qJ = 1/1 o!c. and we use Equation (13). We abbreviate withf~ = Zll + ~Il + ,Il. Then (13) becomes 0~{J

+ 0, + L I/I~(f~)(dz). + oe· + of).) }.

=

yt [!5~

+ Oy ~{J + or ,P + ~ I/I~(h)(ay e' + i3y fA)] qJY.

Multiplying by the inverse of the expression in brackets [- ] we get qJY = o~Y

+

L I/IHf~) dz

A

+ ...

A

= o~y

+ 2: I/IHz) dz}. + RY(I/I, O. }.

Thus,

1/1 f~ = qJ = o~ + 1/1 + R( 1/1, ~), 0

(17)

where R(tl/l, t~) = t2Rl(I/I,~, t) if t is a real number and both R, RI are COO functions of the parameters 1/I'P(z), 1/I'P(!c.(z», ~1l(Z), (a~lllozp)(z), (iJ~lllozP)(z) in local coordinates. In AO we have RO, the space of holomorphic vector fields on M. Using the L2 inner product on AO we let FO be the orthogonal complement of RO. So E F ° if and only if (~, '1) = 0 for all '1 E RO, that is, F ° is the kernel of the map R: AO --+ RO. Then for ~ E FO,

e

Since 8 is zero on A 0, 8~ = 0, and

yielding ~

=

G8o~.

Now give AO, Al and their subspaces the

II Ilk topology.

(I 8)

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

172

PROPOSITION 3.4. There are neighborhoods of the origin V and V in At and F 0, respectively, so that for any 1/1 E V there is a unique = 1/1) in V such that (19)

e e(

Proof

Using (17) we see (19) is satisfied if and only if

0= 8(1/1 By (18)

e=

0

h) =

Me + 81/1 + 8R(I/I, e)·

G8o~ = -G81/1 - G8R(I/I, ~).

Thus (19) is equivalent to

+ G81/1 + G8R(I/I, ~) = O.

~

(20)

Now choosing neighborhoods VI' VI so that R is defined on VI x VI we can define a map h: VI x VI -+ F O by h(I/I, e) = ~ + G81/1 + G8R(I/I, ~). By our previous remarks on R(I/I, e), we see that h is continuous if VI' VI, F o have the I Ilk topology, since R is continuous as a map from VI x VI with the II Ilk topology to A 1 with the II IIk-l topology. In fact, h is even uniformly continuous and hence has a (unique) extension to a mapping h of the completion of the domain to the completion of FO. Then the partial derivative -8h

8~

I ..F---+r, -0

1"0

(0,0)

where po is the completion of F O, is the identity map. Then by the implicit function theorem [see Lang (1962)] there is a COO function g on a small neighborhood of the origin in the space Al (completion of Al)such that Equation (20) is satisfied if and only if = g(I/I) for some 1/1 E U. Thus given 1/1 E V, where V is a small neighborhood in AI, there is a unique solution g(I/I) = ~ of (20) which is sufficiently small. Then 0 + 8R(I/I, ) + 8 is an elliptic secondorder equation with coo. Thus, if 1/1 E V s;;; AI, = g(I/I) satisfies

e

e

oe + 8R(I/I, ~) + 81/1 = 0 so ~ is Coo, that is,

e

E

FO.

Q.E.D.

Let us summarize our conclusions. THEOREM 3.1. (Kuranishi) (a) Let M be a given compact complex manifold, and let {'7.} be a base for W ~ Hl(M, 0). Let qJ(t) be the solution of the equation

qJ(t)

= '7(t) + !8G[qJ(t), qJ(t)],

4. where 11(t) = L~= 1 t. 11.,

STABILITY THEOREMS

ItI < p,

173

and let

OJ.

B = {t I H[ep(t), ep(t)] =

Then for each t E B, ep(t) determines a complex structure M t on M. (b) Let r/J be any vector (0, I)-form satisfying or/J - 1[r/J, r/JJ = O. Then r/J defines a complex structure M", on M. If II r/J Ilk is small enough, there is a unique E F O such that r/J 0 f~ = ep(t) for some t E B, and hence M", is biholomorphically equivalent to M,.

e

4.

Stability Theorems

The main point of this section is to prove that if .,I( = {M,} is a complex analytic family and Mto is Kahler, then M t is Kahler if It - tol is small. We first study elliptic differential equations depending on a parameter. Let P = {tlltl < IX, t = (t1' ... , tr )} be an open disk in cr. Let X be a compact differentiable manifold, and let PA be a differentiable complex vector bundle over X x P. Let B t = PAIXX{tl be the restriction of PA to X x {t}. Then we can consider PA = {B,} to be a family of vector bundles over X depending differentiably on t E P. Let L(PA) = the space of differentiable sections of ffI, L(B,) = the space of differentiable sections of B, . We are only interested in small deformations so we can assume that P is small and then there will be a finite covering {XJ of X such that

PA Ix,xp =CI'

X

Xi

X

P,

that is, PA is trivial over Xi x P. Let (Cf, x, t) be a local coordinate on ffllx, x p. Then the coordinate transformations on PA are written as I'

Ct = •L=1 b!.(x, t)'~ . By an (even-order) differential operator E, : L(B,) -

L(B,),

we mean a map which can be written locally in the form I'

(E,r/J)t(x) =

L Et.(x, t, DMi;(x),

.=

1

where r/J(x) = (r/Jt<x» is a section of B, and where E~(x, t, D i ) is a polynomial of degree m =0 (mod 2) in Di = (a/ax'f). In our applications we will only need m = 2,4. For E, to be well defined we must have

L Et(x, t, Di)bikt(x, t)r/Ji.(x) = L bi1.(x, t)E;.(x, t, D,Jr/Jk(X),

"

.

"

.

174

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

where It

I/It(x) =

L1btkY(X, t)I/IZ(X).

y=

We say that E, depends differentiably on t if all of the coefficients of Ety(x, t,Dj)as a polynomial in Djare C" functions of (x, t). We assume given a Riemannian metric on M and an Hermitian metric on the fibres of B, so that

L gilv(X,t) ,tc dX A,

j

=

\I

L gkClt(X, t)C~ ,r dX k

tI, 't

is invariantly defined. We have written these expressions in terms of local coordinates for X where

,t = L Mix, t),: CI

and dX j

=

dxl'" dx~

(n = dim X).

We assume that gjAv(X, t) are Coo in (x, t). An inner product on L(B,) is defined by (1)

We will consider only those E, which are formally self-adjoint; that is, (E, cp,

1/1), =

(cp, E,I/I),

(2)

for all cp, 1/1 E L(B,). Let us see what this implies for the coefficients of E,. Let E;,;A(X, t, D j) be the terms of order m in Ej~(x, t, DJ Writing out (2) we get

fx L gjAtE'!'/(x, t, Dj)cpax)I/IRx) + lower order terms = f L gmCPt(x)E?;Y(x, t, D;)I/IXx) + lower order terms. x A,t,Y

A,

Y, t

Integrating the first term of the left-hand side by parts and using the fact that m is even we get

This is true for all cp and 1/1 so we get

L gjAvE~Yl/li = L gjYtE';/I/Ii Y, t

for all 1/1. Hence, (3)

4.

STABILITY THEOREMS

175

Replace (a/ax~) in Er';,v with YIZ' Then Er;,V(x, t, y) is a homogeneous polynomial of order m in y with coefficients which are Coo functions of (x, t). Equation (3) becomes

if Aw = "Iv giv, Er;,V(x, t, y). We have proved: PROPOSITION 4.1.

Equation (2) implies (3).

We shall assume further that Er is strongly elliptic, that is, (_l)m/2 "I Ai).t(x, t, y)w A W, > 0 A.

t

for any real y = (Yl' "', yft) =F 0, and any complex W = (WI' " ' , w,,) =F O. We need to collect some facts about such operators. We quote the following well-known theorem which can be found, for example, in Palais(1965, p. 182): THEOREM 4.1. Er has a complete orthonormal set of eignenfunctions {erh}h": I ~ L(Bt )· Let the eigenvalues be Ah(t). Then they are real and with (erh' er)r = b jk • [Completeness means that any 1/1

E

L(B r) can be written as follows:

00

1/1

=

"I ajeth , h=l

Furthermore we can arrange the e th such that

and Iim h -> 00 Ah(t) =

+ 00.

The following theorem is proved in Kodaira and Spencer (1960), and we shall not prove it here. THEOREM 4.2. REMARK.

Each eigenvalue Ah(t) is a continuous function of t E P.

Ah(t) may not be differentiable. For example, let E = ((X(t)P(t») t

y(t)b(t)

be a Oth-order differential operator (just a matrix). Then 1)

AU =

(X(t) + c5(t)

± J«(X(t) -

b(t»2

+ 4p(t) b(t)

-'--'----=---~~..:----..:---=-----=----=-----=--

2

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

176

which could fail to be differentiable when (ex - (j)2 the kernel of E, . Then

1Ft

+ 4fJ(j =

O. Let 1Ft be

= {I/I IEt 1/1 = O} = {I/I I1/1 = )..(1)=0 L ah eth}.

Let F, be the orthogonal projection to 1F t , that is,

F,I/I =

L

(1/1, e'h)e'h·

)..(1)=0

The Green's operator G, is defined by

G,I/I=

L

1

1 (t)(I/I,elh)e,h .

)..(I)"'O/Lh

F, and G,are related by the equation

1/1 =E,G,I/I

+ F,I/I.

We have already investigated the case P = a point, E, = D, F, G, = G. In the general case we have the following theorem:

= H,

IFr

= IHI,

THEOREM 4.3. dim IF, is an upper semicontinuous function of t. This means that given to , there is a small enough e so that dim IF I ~ dim IF to for It - tol < e. Proof dim IF, = d, is finite since An(t) -+ Ah(tO) we have

...

~

00.

In the ordering of the

Aito) < 0 = Aj+ I (to) = ... = Aj+do(tO) < Aj +do +I(t)

~

By continuity, choose e so that Ait) < 0 and Aj+do+l(t) > 0 if Then in this disk d, ~ do. Q.E.D.

....

It -

tol <

&.

Next we want to say what we can about the differentiability of F, and Gr. DEFINITION 4. I. Given 1/1, E L(B,) for each t E P, we say that 1/1, depends differentiably on t if there is a 1/1 E L(fA) such that

1/1,

=

I/Ilxx(I}·

Let (ej. x,,) be a local coordinate on fA, where

e; = L b;kv(X, t)ez· Then if 1/1, = {I/I~(x)} in local coordinates, we have I/I~(x) =

L b;k.(X, t)I/I:k(X).

If 1/1 E L(fJl) is such that 1/1, = 1/11 x x (I)' then 1/1 = {I/I1(x, t)} and 1/11(x, t) = I/I:ix) so I/I~(x) are Coo in x and t. Conversely, if the I/I:ix) are Coo in x and I, it is easy to see that 1/1, depends differentiably on t.

4.

177

STABILITY THEOREMS

DEFINITION 4.2. Given a linear operator A, : L(B,) -+ L(B,) for each t E P, we say that A, depends differentiably on t if the following condition is satisfied: If 1/1, depends differentiably on t, then A, 1/1, depends differentiably on t. REMARK.

E, depends differentiably on t in this sense.

Given E" let I = [a, fJ] be a real interval on the .A.-line such that 0 E 1. We define

THEOREM 4.4. Let to E P. If .A.h(tO) =!- ce, Pfor all h, then there is an open set V around to such that F,(1) and G,(J) depend differentiably on t E V.

Sketch of Proof

Consider a rectangular contour C in the (-plane as in

the figure below.

r

c a

Ii

A-line

Figure 10

Let'

E

C and form E, - (. Then we have

II(E, - ()I/III; =

ilL (1/1, e,h)(Ait) - Oe,hll:

~ min /Ah(t) _ ,/2111/111~ h

where k is some constant (greater than zero), as long as t remains in a small compact neighborhood of to. This implies easily that (E, - 0- 1 = G,(O is defined and continuous. It is harder to see that G,«() depends differentiably on (t, (). This is proved in Kodaira and Spencer (I 960). We can write down GrC() :

178

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Then integrating around the contour C we get

1. F,(l) = - -2 1tl

JCG,(O d,

because

If

d,

2ni c A. - , =

{-1,0,

iOEI

irA. ¢ I

(A is real).

We also have

since

Thus G,(I) and F,(I) depend differentiably on t.

Q.E.D.

4.5. If dim IF, is independent of t E V, where V is some open subset of P, then F, and G, depend differentiably on t.

THEOREM

Proof Since the Ah(t) are continuous in t and dim IF, is independent of t, we can find a small open set U around t and small interval 1= [ -e, +eJ around 0 such that F. = F.(I) and G. = G.(I) for S E U. Then apply Theorem 4.4. Before we apply these theorems, we need to make some more definitions. 4.3. By a differentiable family of compact complex manifolds we mean a triple (.It, iii, P) where ..,II, P are differentiable manifolds, iii is a differentiable map of rank = dim P, and each M, = iii-I(t) is a complex manifold. More precisely for each Xo E .It there should be a local diffeomorphism from IlI!j 3 Xo DEFINITION

X---+ (Zl , •••

, z" , t l , •.. , tm)

to a domain in e" x IR m, where iii(x) = (tl, .. " tm) are coordinates around w(x), and such that for a fixed t, (Zl(X), ... , z"(x» is a complex coordinate on M,. On iJiJ j n iJiJ k'

where jj~ is differentiable in (Zk' t) and holomorphic in Zk for a fixed t. Let T(.It) be the complex vector bundle over .It whose transition functions on

4.

179

STABILITY THEOREMS

iJlt j n UU k are given by the matrix

(~;D· T(Jt) is the bundle of holomorphic tangent l'ectors along the fibres of Jt. Let T*(.,I() be its dual and T(A) its conjugate bundle. Then we set ff*(r, s) =

(~

T*(A»)

(~

A

f*(A»).

We call ff*(r, s) the bundle of (r, s)-forms along the fibres. Let r, be restriction to M,. Then we have r,(T(A» = T(M,)

and r,(ff *(r, s) = T *(r,s)(M,).

Let L r ,. be the space of COO sections of ff* (r, s). Then L~" = r,L r ,. is the space of Coo (r, s)-forms on M,. Any'" E L r , . can be written 1

'" = r! s! L "'a,·"a. fl,···P.(z, t) dzj'

dzj'

A ... A

A

dz~'

A .•. A

dzP"

where dzj = (OIozj)*. Our transformation law for (%zj) is

0)- I -oz~J( -0) (oz% - P~ oz'k oz~ . n

1

Thus the law for dzj is " az«' _ J dz P dz«'J = '" ~ a k' P~ 1

IZ

Zk

This implies that dzj is not the differential of zj. If it were, it would transform according to the law n

dzj =

oz«'

m

az«'

I ~ ~ dzf + )'~l I ul~ i dt),. /l~1 uZk

Next, let fII be a vector bundle on A and let B, = r,(fII). Then L r , '(fII) is the space of Coo sections of fII ® f7*(r, s) and L r , S(B,) is the space of COO sections of B, ® T,*(r. s)(M,). If fII is given by the transition equation II

C7 = L b7kv(Z, 1)C~ v~

and",

E

1

L T , S(81) is given locally by '"

= ('" J' ... , I/I,}), then

180

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

where 1

./,A _ _ '" ./,A 'I'j I I ~ 'I'jcz,· ..

r.s.

Ps /I,···/I,( z, t) d Zjcz, 1\ .•. 1\ dZ .

We have assumed an Hermitian metric on PA depending differentiably on t; hence, if CPt> t/J, are in L(B,) and are COO in t, then (cp" t/J,), depends differentiably on t. We can introduce the operators d" 0" 0" 8" lJ" [}., acting on L~" = L" 'CB,). Then the operators

0, = 0,8, + 8,0" D., = d,lJ, + lJ,d" 0, depend differentiably on t. We now state the main theorem of this section. THEOREM 4.6. Let.A ~ P be a differentiable family. If M,o = or l(t O) is a Kahler manifold, then M, = in-I(t) is Kahler for It - tol small enough. REMARK I. A good problem would be to find an elementary proof (for example, using power series methods). Our proof uses nontrivial results from partial differential equations (Theorem 4.1). REMARK 2. Hironaka (1962) has given an example of a non-Kahler deformation of a Kahler manifold. Hence the theorem is only true for It - tol small.

Proof (of the theorem) Assume P = {tlltl < I} s; IRm. Since Mo is Kahler we have a Kahler form Wo on M 0 Wo =

iL gcz/l(z) dzcz 1\ dz P.

We extend this to an Hermitian metric on all of the fibres as follows: Let .. " tm) be local coordinates on o/J j ~ Uj x P s; en x IR m, such that W(ZI.J" ... tm) = (t 1" ••• tm) Then on o/J. . J

(z}, .. " zj, tl,

is a Coo Hermitian form which is independent of t. Let {piz, t)} be a partition of unity subordinate to {Olt j }. Then we define

w, =

L Pj(z, t)Wj(z) = i L g,cz/l(z) dzcz 1\ dzP. j

cz,p

The metric w, depends differentiably on t and Wo is the Wo we started with. Let our inner products (cp, t/J), be defined with respect to the metric W, where cp, t/J E L~·". Then the operator 0, = 8, + 0,8, is strongly elliptic and depends differentiably on t so our theorems apply. In fact, one can check that the principal part of 0 I is

a,

4.

STABILITY THEOREMS

181

in a local coordinate where (gflJ) is the inverse matrix to (grlJII) (see, for example, Chapter 3, Section 2). We define the following spaces:

Za;s =

{lp I lp

Z3;s =

{lp I lp E IJiS, a, lp = O},

E L~'so,

lp = O},

Lq "L.. L'" t -r' r+s=q

= {lp I lp E L~, d,lp = O}, ~~.s = {lp I lp = L~'" 0, lp = O}, z~,

~~

= {lp I lp E L;, A,lp = O}.

The theorems of Dolbeault and de Rham yield

zr..·it,

S

n..nr,' '"

LflI t

=

3

r, s

u,L,

1

nr) , = HS(M t' .l"t

'"

(4)

~i ~ W(M" C),

where

n~

is the sheaf of germs of holomorphic r-forms on M,. As usual, let

h:"

= dim ~~", bq = dim ~i.

Then bq is the qth Betti number of M, (which is independent of t since all of the M, are diffeomorphic to each other). Mo is Kahler, so

6. 0 = 20 0 = 200'

(5)

This was proved in Chapter 3, Section 5. We have for each t

0,0, + 0,0, = 0 = 9,0, + 9,9"

(6)

since Kahler is not necessary for this. However, if the reader will consult the proof of (5) he will find that in the Kahler case (for example, t = 0) (7) Now we define

E, = 0,0,9,9, + 9,9, a, 0, + 9, a, 9, a, + 9, a, 9, a, + 9/0, + 9, a, . Then

(E,({J, 1/1), = (9, 0, lp, 9,9,1/1), + (o,O,lp, 0,0,1/1),

+ (9,u,lp, 9,0,1/1), + (o,lp, 0,1/1), + (o,lp, 0,1/1),. PROPOSITION 4.2. E, is a strongly elliptic self-adjoint differential operator of order 4 acting on L~·s.

Proof

We clearly have

(E,lp, 1/1), = (lp, E,I/I),.

(8)

182

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Thus E, is self-adjoint. We let the reader check that the principal part of E, is 4

" fJ« 6y 0 L.. gi gi ~«~ p ~ y ~-6 «, p, y, 6 UZi UZi UZi UZi

in a local coordinate system. PROPOSITION

8,9,cp

=

o,cp

Proof

Q.E.D.

4.3. Eo = Do Do = a,cp = o.

+ 8000 + 9000 and

E, cp = 0 if and only if

Using Equations (5), (6), and (7) we get

Do Do

= (9 0 00 + 00 90)(8 0 00 + ( 0 80) =

90 ao80 00 + ao90 00 80 + 00 9 0 90 00 + 90 00 Vo 8 0

= 80

90 ao 00 + 00 00 8 0 90 + 8 0 ao 90 00 + 90 00 80 00 .

This proves the first statement. For the second statement we use the obvious inequality Q.E.D. As before, let 1F~'s = {cp I E, cp = 0, cp E L~·S}. Let F,: L~'s -+ orthogonal projection to 1F~'s and G, be the Green's operator, so

1F~'s

be

t/! = E,G,t/! + F,t/!. PROPOSITION

4.4.

zr,s ,s-l t.:p d, -- a, 0, L,-1 , w F.r,s , , where t.:p w means orthogonal

direct sum. Proof If cpEIF~'" then d,cp=O. If t/!Ea,o,L~-l.s-1 then, d,cp=O sinced,o,o, = (a, + o,)a, 0, =0. Thus a,o,L~-1.s-1 + 1F~'ss;Z~;·. The sum is orthogonal since

(a, a, cp, 1]), = (cp, 9,9,1]), = 0 for 1] E 1F~'s. Next take

t/! E Z~·s.

Then

~I = E,G,t/!

+ F,t/!

= a, 0, rx

+ 9,f3 + 9,y + '1.

where 1] = F, t/!. Since d, t/! = 0

d,(8,P + 9,y) = Let 0'=8,p+9,y. ThenO'EL~'s and a,O' = 0,0' = O. We claim 0' = O. For

(0',0'),

o.

a,:L~'·-+L~+l.S,o,:L~,s-+L~,S+l,

= (8,P + 9,y, 0'), = (P, 0,0'), + (y, 0,0'), = o.

(9)

so

4. Thus u = O. Hence

t/I

=

183

STABILITY THEOREMS

0,0, c< + '1, and Q.E.D.

dim IF}·1 ~ b 2

LEMMA 4.1.

2h~·2 where b 2 is the second Betti number of

-

MI' Proof O IF ,I • 1 -~ ZI.I/ 0"0 ,L. d,

Since

o, 0, LO, - dL ,

I

C

ZI.1 n ZI,I ) dim IF I.1 > dim ( _d_, ,dL ,I d,

n ZI.l d,

+ £ILl) = dim (ZI,I d, '. d,LI,

By de Rham's theorem b 2 = dim (Zi,/d,L,I). We claim there is the following exact seq uence ZI.1 + £I Ll, d" d, Ll, -

o

-

Z 2" Z2.0 Z!l·2 d,' ill D, 1 •O + aLo. 1 · d, L ,1 - o L " "

We must define 1t, and check that it has the correct kernel. Let t/I E zi" d, t/I = t/l2.0 + t/ll.l + t/l0.2. Then d,t/I = 0 yields o,t/l°·2 = o,t/l2.0 = O. So we can map ./, to ./,2.0 + ./,0.2 E Z2.0 + Z!l·2 Let ./, Ed L 1 Then 'I' 'I' 'I' 0, D,' 'I' ".

o where t/I =

t/I

= d,(cpl,O

+ cpO.l)_ O,cpl.O + O,cpl.O.

This correspondence induces the map 1.O+ 0,.°·1. Then t/l2.0 + t/l0.2

=o,u

t/I where

Thus

~

e

d,(u 1•O + .0.1) =

~ =

'1t,. To compute ker 1t"

t/ll.l _ o,u 1•0

_ 0,.0.1

suppose 1t, t/I =

= ~1.1

is of type (l, 1) Then

E

ZJ;1 and this yields

b

2 -

dim(ZJ;1

d,e = d,t/I = O. t/I E d,L,1 + ZJ;l.

This exact sequence implies

;~;L:) ~ dim(o~tl~o) + dim(o~t;l)'

But Dolbeault's theorem implies dim

Z(P) (u,L, :os

1"0.1

= dim H2(M, l!J) = ,,~.2

and

hO•2 " Thus dim 1F,1.1 ~ b 2 - 2h~·2.

Z2.0 ) = h2 •o = dim ( _0_,_ I o

0, L , • .

Q.E.D.

184

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

LEMMA 4.2. Proof

dim IFf,l = dim IFA'o for smalliti. Let us compute dim IFA,l. We claim 1F~,1 = Il~,l ~

HI(Mo, (

1).

(10)

We use (5) and Proposition 4.3 for the proof of (10). Since

Eo cp = 0 implies Do cp = O. Conversely if Do cP = 0, then 00 cP = 0 = 8 0 cP; and Do cP = 0 so o cP = 90 cP = O. Thus Eo cP = O. This proves (10). Hence,

a

Recall that on a Kahler manifold

b2 = =

+ hA,l + hg,2 h~,l + 2hg,2. h~'o

Thus, dim

1F~,1

= b 2 - 2hg,2.

By the upper semicontinuity of dim 1F,I,1, for It I <

8.

Also by upper semicontinuity of dim 1HI?,2, for

It I < 8.

Thus, dim 1F~,1 ~ dim 1F1,l ~ b2 COROLLARY. Proof

-

2h~,2 ~

b 2 - hg,2 = dim

1F~,l.

Q.E.D.

F, depends differentiably on t.

Use Proposition 4.5.

Now we shall finish the proof of Theorem 4.6. We have

w, = i

L 9«I1(Z, t) dz« 1\ dzfJ ,

«,fJ

which depends differentiably on t and Wo is a Kahler form Mo. Hence, dwo = O. In fact, if the reader will consult Chapter 3, Section 5 he will find

4.

STABILITY THEOREMS

185

we proved that COo E 1HI~,1 = IFA,I. Thus Focoo = COo. Since 1/1, = F,co, depends differentiably on t, 1/1, -+ COo as t -+ O. The form CO, is of type (1, 1) and satisfies a, t/I, = 0, t/I, = 0 so dt/l, = O. Let

w, = t(l/I, + iii,). Then w, is a closed (l, 1)-form which is positive definite for small t since Wo = COo is positive definite. Thus W, is a Kahler form on M, for small Itl. Q.E.D.

Bibliography Adler, A., The second fundamental forms of S6 and i?"(C). Amer. J. Math., 91, 657-570 (1969). Benzecri, J. P., Varietes localement affines. Seminaire Ehresmann (May 1959). Borel, A., and J.-P. Serre, Groupes de Lie et puissances reduits de Steenrod. Amer. J. Math., 75, 409-448 (1953). Bott, R., Homogeneous vector bundles. AI/n. Math., 66, 203-248 (1957). Calabi, E., and E. Vesentini, On compact, locally symmetric Ktihler manifolds. Ann. Math., 71, 472-507 (1960). Dieudonne, J., Foundations of Modern Analysis. New York: Academic Press, Inc., 1960. Douglis, A., and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math., 8, 503-538 (1955). Ehresmann, C, Sur les espaces fibres differentiables. c. R. A cad. Sci. Paris, 224, 1611-1612 (1947). Fischer, W., and H. Grauert, Lokal-triviale Familien kompacter komplexen Mannigfaltigkeiten. Nachr. Akad. Wiss. Gottingen Math-Phys. KI 11,89-94 (1965). Frolicher, A., and A. Nijenhuis, A theorem on stability of complex structures. Proc. Nat. A cad. Sci. U.S.A., 43, 239-241 (1957). Gunning, R. C, and H. Rossi, Analytic Functions of Sel'eral Complex Variables. Englewood Cliffs, N. J.: Prentice Hall, Inc., 1965. Helgason, S., Differential Geometry and Symmetric Spaces. New York: Academic Press, Inc., 1962. Hironaka, H. An example of a non-Kahlerian complex analytic deformation of Kahlerian complex structures. AI/n. Math., 75, 190-208 (1962). Hirzebruch, F., Neue Topologische Methoden ill der algebraische Geometrie. 2d edition. Berlin: Springer-Verlag 1962. ---,Uber eine Klasse von einfachzusammenhangenden komplexen Mannigfaltigkeiten. Math. Ann., 124, 77-86 (1951). Hirzebruch, F., and K. Kodaira, On the complex projective spaces. J. Math. pure et appl., 36,201-216 (1957). Hbrmander, L., Linear Partial Differential Operators. Berlin : Springer-Verlag, 1963. Kodaira, K., On the structure of compact complex analytic surfaces I. Amer. J. Math., 86, 751-798 (J964). --,On cohomology groups of some compact analytic vanetie<; with coefficients in some analytic faisceaux. Pro£'. Nat. A cad. Sci. U.S.A., 39, 865-868 (1953). ---,On Kahler varieties of restricted type. Ann. Math., 60, 28-48 (1954). ---,On compact complex analytic surfaces I. 1/11/. Math., 71, 111-152 (1960).

186

BIBLIOGRAPHY

187

Kodaira, K., L. Nirenberg, and D. C. Spencer, On the existence of deformations of complex analytic structures. Ann. Math., 68, 450-459 (1958). Kodaira, K., and D. C. Spencer, Groups of complex line bundles over compact Kahler varieties. Proc. Nat. Acad. Sci. U.S.A., 39, 868-872 (1953). ---,On deformations of complex analytic structures, I and II. Ann. Math., 67, 328-466 (l958a). ---,A theorem of completeness for complex analytic fibre spaces. Acta. Math., 100,281-294 (195Rb). ---,On deformations of complex analytic structures, III. Stability theorems for complex structures. Ann. Math., 71, 43-76 (1960). Kuranishi, M., New proof for the existence of locally complete families of complex structures. Proceedings of the Conference 011 Complex Analysis in Minlleapolis 1964. Berlin: Springer-Verlag, 1965, pp. 142-154. Lang, S., Introductioll to Diflerellfiable Manifolds. New York: Interscience Publishers, 1962. Milnor, J., Morse Theory. Princeton, N.J.: Princeton University Press, 1963. Nakano, S., On complex analytic vector bundles. J. Math. Soc. Japan, 7, 1-2 (1955). Newlander, A., and L. Nirenberg, Complex analytic coordinates in almost complex manifolds. Anll. Math., 65, 391-404 (1957). Palais, R. S., et al., Seminar on the Atiyah-Singer Index Theorem. Annals of Mathematics Studies No. 57. Princeton, N.J.: Princeton UniverSity Press, 1965. Serre, J.-P., Une theoreme de dualite. Comm. Math. HelL'., 29, 9-26 (1955). Teichmtiller, 0., Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. der Wiss. Math.-natllrw. Klasse, 22, 1-197 (1940). Wu, W. T., Sur les classes caracteristiques des structures fibrees spheriques. Actualites Sci. Indust., Herman/l, Paris (1952).

Index

A

Algebraic, projective, 11

Dolbeault's lemma, 74, 79 Doibeauit's theorem, 80 Dual form, 93

B

Blowing up, 17 Bounded domain, 144 Bundle, vector, 62 conjugate, 65 dual, 65 quotient, 66 section, 66 sub,66 tensor product, 65 Whitney sum, 65

F

Family, complex analytic, 18 differentiable, 178 Fine resolution, 73 Form, differential, 67 G

Germ, function, 27 Green's operator, 158, 176 Group, discontinuous, 12

C H

Cauchy-Riemann equations, 3 Chern class, 64, 127 Completeness, 55 theorem of, 56 Conjugate tangent bundle, 66 Coordinate, local complex, 7 Covariant differentiation, 106-109 Curvature, Ricci, 118 tensor, 117

Hartogs' lemma, 24 Hermitian metric, 83 Hirzebruch, F., 15 Hodge decomposition theorem, 114 Hodge metric, 134 Holomorphic, function, 1 mapping, 10 I

D

Deformation, 19 infinitesimal, 36-39 small, 19 de Rham's theorem, 72 Derivative, exterior, 68 Differentiable mapping, 10 Differential operator, 173

Implicit mapping theorem, 7 Integral, 88 Inverse mapping theorem, 6 J

Jacobian,S

191

192

INDEX

K Kiihler, form, 84 manifold, 84 metric, 84 Kodaira, K., embedding theorem, 136 vanishing theorem, 125, 131 Kuranishi, M., theorem of completeness, 172 L

Laplacian, 97 Locally trivial, 39 Logarithmic transformation, 16

M Manifold, complex, 7 Hodge, 134 Hopf,14 Kiihler, 84 Meromorphic function, 11

N Nakano, S., vanishing theorem, 132 Newlander-Nirenberg theorem, 156 Norm, 92 Sobolev, 166

o Oriented, 88 Osgood's theorem, 2

P Partition of unity, 61 Poincare's lemma, 70 Polycylinder,2 Polydisc,2 Positive line bundle, 131 Product, inner, 92 wedge, 67

Projective algebraic, 11 Pseudogroup, 8

Q Quadratic differentials, 106 Quadric transformation, 17 Quotient space, 12

R Refinement, 31 Rigid,45

s Semicontinuity, upper, 45 Serre duality, 104 Sheaf,29 cohomology, 31 exact sequence, 57 fine, 61 homomorphism, 56 quotient, 57 section, 29 Sobelev lemma, 166 Stability of Kiihler manifolds, 180 Stalk,29 Stokes' theorem, 91 Structure, canonical, 9 complex, 7 flat affine, 9 Submanifold, 11 Subsheaf, 56 Surface, Hopf, 23 ruled, 15, 25, 41 Surgery, 15 T

Tangent bundle, 66 Tensor bundle, 67 Torus, 13, 21-23

v Vector field (holomorphic), 36

Errata Page v, line 9: "lines fundles" should read "line bundles". Page 1, line -9: Ck ... k n should read Ck 1 ••• k n • Page 2, line 1 after the figure: There should be a comma between and i. Page 2, line -5: ~ should read ~. Page 3, lines 9, 11, and 13: Ckl .•• kn should read Ckl ... k n • Page 3, line -3: az" should read az". Page 4, line -13: i ~ 1/ ~ n should read 1 ~ Page 7, line -15: zU(P) should read zn(p).

1/

~

IWi I

n.

Page 10, line 16: ~~~tl x~ should read ~~~tl x~ = 1. Page 10, line 21: v = 1,2 should read 1/ = 1,2. Page 12, line 12: pI should read pl. Page Page Page Page Page Page

12, 13, 13, 18, 19, 26,

line -5: pEG should read 9 E G. line 2: Fm+s should be Fm+!. line 2: gm should read g, twice. line 10: 1m w -+ 0 should read 1m w > 0 . first line of the diagram: should be ~. line 5: z(f should read

!

zr.

Page 32, line 13: 71'1 ... l'q should be replaced with (kT)I'I ... l'q' Page 33, line -9: Omit the first "f/ (in the subscript of II). Replace the second script "f/ with %' . Page 33, line -5: Replace the first occurrence of II~ with II~. Page 35, line 11: Replace all occurrences of Y with (j Page 38, line 5: 11>'6 should be 11>.". Page Page Page Page Page Page

39, 41, 42, 43, 44, 55:

line -10: Mo should be Mb. lines 3 and 4: ~ 1 should be> 1. line 10: Zl should be Z2. line -6: aZ l should be a(l. line 10: -mglOzlblO should be blO - mglOZl. Add the map q> to the top arrow of the diagram. 193

ERRATA

Page 58, lines -13 and -6: E cO should be E Co. Page 61, line 3 after the diagram: ht.p should be kt.p. Page 64, line -13: n 2:: Z should read n 2:: 2. Page 64, line -6: hk(Z) = should read fik(Z) =. Page 80, line -3: There should be a space after the comma between t.p and q. Page 81, lines 12 and 13: There should be a comma after OJ "The" should be "the". Page 85, line -11: The last term in this equation should read 8 f L:Q,,B ~dza A dz,B. 2

Page 85, line -2: dZ a should read dz a . Page 86, line 10: 2 L: 9ja,B should read 2 L: 9jaij· Page 87, line -4: The second z in this equation is missing a subscript j. Page 93, line -13: '!/J ApB• should be '!/JApB. j JL1 should be JLI. Page 93, line -9: Aq should be Ap both times. Page 93, line -7: The first B in this equation should NOT have a bar. Page 95, line -5: t.p A *'!/J should read t.p A *i[;. Page 95, last line: The left-hand side of the last equation on this line should read o'!/J = instead of 'I3'!/J. Page 100, line -10: The second equation on this line should read - A 8f jk v(z) = 0. Page 108, line 12: omit the bar over 8>.. Page 112, line -7: Proposition 5.4 should read "In the Kahler case

"

Page 118, line 5: The last factor in the subscript of the R in the right-hand side of the equation should be Y. Page 118, line 9: The subscript on the last R should be {3v),.. Page 120, line -4: {31 should be i31' Page 120, last line: (r)i should be (r)k. Page 124, line 10: x should be +. Page 126: line 5: {3q should be Bqj the term involving t.p after the summation sign should be t.pjB• • t.p: •• Page 126. lines 11, 12, and 13: All superscripts Bq should be Bq; in line 13 there should be a bar over the entire expression Vat.p:·.

Complex Manifolds (AMS Chelsea Publishing)

Read more

Finite Groups (AMS Chelsea Publishing)

Read more

Almost Periodic Functions (Ams Chelsea Publishing)

Read more

Knots and Links (AMS Chelsea Publishing)

Read more

Calculus of Finite Differences (AMS Chelsea Publishing)

Read more

Class Field Theory, Second Edition (Ams Chelsea Publishing)

Read more

Conformal Invariants: Topics in Geometric Function Theory (Ams Chelsea Publishing)

Read more

Analytic Function Theory, Volume I - 2nd Edition (AMS Chelsea Publishing)

Read more

Einfuhrung in die kombinatorische Topologie (AMS Chelsea Publishing)

Read more

Selected Topics in Algebraic Geometry, Second Edition (AMS Chelsea Publishing)

Read more

Conformal Invariants: Topics in Geometric Function Theory (Ams Chelsea Publishing)

Read more

Class Field Theory, Second Edition (Ams Chelsea Publishing)

Read more

A Course in Ring Theory (AMS Chelsea Publishing)

Read more

Theory of Functions of a Complex Variable, Volume 1 (Ams Chelsea Publishing No 97)

Read more

Complex Manifolds

Read more

Arithmetic of complex manifolds

Read more

Lectures on complex manifolds

Read more

Complex manifolds and deformation of complex structures

Read more

Complex Manifolds and Deformation of Complex Structures

Read more

Essay d'analyse sur les jeux de hazard (AMS Chelsea Publishing) (French Edition)

Read more

Several complex variables and complex manifolds II

Read more

Solvable Models In Quantum Mechanics With Appendix Written By Pavel Exner (AMS Chelsea Publishing)

Read more

Vanishing Theorems on Complex Manifolds

Read more

Differential Analysis on Complex Manifolds

Read more

Chelsea Chelsea Bang Bang

Read more

Chelsea Chelsea Bang Bang

Read more

Chelsea Chelsea Bang Bang

Read more

复流形(Complex Manifolds)

Read more

Differential Analysis on Complex Manifolds

Read more

Differential Analysis on Complex Manifolds

Read more

Recommend Documents

Complex Manifolds (AMS Chelsea Publishing)

COMPLEX MANIFOLDS JAMES MORROW KUNIHIKO I(ODAIRA AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rh...

Finite Groups (AMS Chelsea Publishing)

Almost Periodic Functions (Ams Chelsea Publishing)

Knots and Links (AMS Chelsea Publishing)

Calculus of Finite Differences (AMS Chelsea Publishing)

CALCULUS OF FINITE DIFFERENCES BY INTRODUCTION BY SECOND EDITION CHELSEA PUBLISHING COMPANY NEW YORK, N.Y. 1950 i...

Class Field Theory, Second Edition (Ams Chelsea Publishing)

CLASS FIELD THEORY EMILARTIN JOHN TATE AMS CHELSEA PUBLISHING American Mathematical Society • Providence:, Rhode Islan...

Conformal Invariants: Topics in Geometric Function Theory (Ams Chelsea Publishing)

CONFORMAL INVARIANTS TOPICS IN GEOMETRIC FUNCTION THEORY LARS V. AHLFORS AMS CHELSEA PUBLISHING American Mathematica...

Analytic Function Theory, Volume I - 2nd Edition (AMS Chelsea Publishing)

Einfuhrung in die kombinatorische Topologie (AMS Chelsea Publishing)

Selected Topics in Algebraic Geometry, Second Edition (AMS Chelsea Publishing)