COMPLEX D ~ G E O M E T R Y

Robert E. Greene

I. Couple . man/foldm

Let C = the complex numbers and C " = C × "" xC (n factors). C" wE]be

referred to as n-dlmensional complex euclidean space. Of course, C" is topologically

2n -dimensional; more specifically, C n can and will be identified homeomorphical]y

with R 2" as follows: If (z1,...,z,) E C , and if z /= x / + [y j ,x j ,y j 6. [g, j = I . . . . n

then (Zl, .... Z,) is tO be identified with (xl,yt, . . . ,x, ,y,) E R 2". That is, (zl,. . . ,z,) is

identified with (Re zt,lm zl , . . . ,Re z , , Im z,), where Re and Im denote real and

imaginary parts, respectively.

DEFINITION. A function f : D -- C defined on an open subset D of C" is

ho/omorph/c if f considered as a function from D C R z" to R 2 is C ® and satisfies

the Cauchy-R/emann equations:

0 ( ~ f ) = 0(Ira f) 0xj 0yj

and O(Ref) = _ 00m f ) ayj Oxj

for all j = l, . . . ,n. A mapping f : D -- C m is ho/o,norp?~¢ if each component of / is

holomorphie; that is, if/1,---~fm are holomorphic, where f / : D -- C, j = I ..... ,I are

defined by/C~) = ~ACz),...J, Cz)) ~ c ~ for ~ ~ z>.

Note that if D = UX ~A DX with each D>, open, then f : D -- C m is holomorphic

if and only if f I D~ is holomorphic for every k.

Multiplication by i, (zt,... ,z,) -. (izl,...,/z,), defines a mapping from C" to C",

whose square is multiplication by -1 . The induced mapping from R 2n to R ~ , which

will be denoted by J, is a real linear endomorphism, whose composition with itself is

again multiplication by - 1 . The following lemma shows that the endomorphism J can

be used to characterize holomorphic mappings:

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L E M M A 1. A mapping f : D -. C m, D c ~ C C n, which is C ® "when considered as

a function from D C R 2n to R 2m i~ holomorphic if and only O e f , JR~ = JRz.f, . Here

f , i~ the real Jacobian of f .

eroot'. Write f (z) = O'~(~),...,fm(~)), fj(~) ~ C for j = 1 . . . . . m. In matrix form

relative to the standard bases of it ~' and R2",

b =

a(~fl) a(~f9 axx ayx

a(Imf 1) a(Imf I) 8xl 8yl

aCR~ m) a(Ref m) c~xx ay 1

aO-,f m) agmf~,) aXl By 1

\ a(Xef~) a(~f~) \

axn ayn

a(T~f m) a(Imf .,)

ax. ay n

a(ncfm) a(R~.) ax. cTy. /

a(Im~ m) a(Imf m) cTxn c~y n

0 -1 1 0

- 1 0

0 1

- 1 0

JR ~

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O -1 1 1 0

0 0 - I 1 0

0 0 -1 / \ o

Straightforward matrix multiplication shows that f , J = J f , if and only if

a(Re f t ) a(Im f t ) a(Im f t ) a(Re fk) = and =

Oxj ayj ax i ayj

for all k = 1 . . . . ,m and j = 1 , . . . ,n .

I.emma 1 implies immediately the following corollaries:

COROIJ~ARY. I f f : D -. C m and g : D 1 ~ C ~ are holornorphic mappings and i f

f ( D ) CD1 °pen C C m, then g o f : D -. C k is holomorphic.

C O R O I J A R Y . I f f : D -. C n, D °pen C C n, is holomorphic and is a diffeornor-

phism onto its image when considered as a mapping f rom D C R 2n to R 2n then

f - 1 : f ( D ) -* C n is holomorphic. (Note that in this case f ( D ) is necessarily open in Cn.)

DEFINITION. A complex structure on a C ® manifold M of even dimension 2n

is a maximal collection of C ® charts indexed by a set A:

{(~x,U~) : x E A , ~ : Ux -. R z~}

having U x (A Ux = M and satisfying the following condition (holomorphic transition)

for all k,tL E A,

~x~b~ 1 : ~b~(Ux F3 U~) -. ~h,(Ux N U~) C a 2n

is a holomorphic function considered as a function from an open subset of C n to C".

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A complex manifold is a paracompact C ® manifold M together with a complex

structure. The number n is the complex dimenaion of M. The coordinate charts of the

complex structure of a complex manifold axe called holomorphic coordinate charts or holo-

marphic coordinate systems.

Note that an open subset of a complex manifold is itself a complex manifold in a

standard fashion.

L E M M A 2. Any collection of C ® charts on a paracompact C ~ manifold M which

cover M and which satisfy the holomorphic transition condition determine one and only one

complex structure on M; i.e. any such collection is contained in a unique maximal such col-

lection.

The proof of this lemma follows exactly the pattern (using the corollaries of Lemma

1) of the standard proof of the corresponding result for real manifolds.

Let M be a complex manifold and p be a point of M. Denote the (real) tangent

space of M at p by Alp. An endomorphism Jp : Mp -. Mp of the real vector space

Mp with jp2 = multiplication by - 1 can be defined as follows: Choose a holo-

morphic coordinate system ~ : U -- R2n defined in a neighborhood U of p. Then

dj,p : Mp - R 2n is an isomorphism (the tangent space to R ~ at d~(p) being identified

with R2~). Jp is then defined to be ~Tp 1 JR~O,p. Lemrna 2 implies immediately that

Jr, thus defined does not depend on the choice of ~b : if ~b is a second holomorphic

coordinate system around p then at p

~,:IJR,.. ~. = ~:~ (,~-l):~.,'R~.(,e~-~).~.

because (~b-1). commutes with JRZ~ by Lcmma 2 and the holomorplicity of ~b -I.

Hence

Thus Jp is independent of holomorphic coordinate choice; that J~ = multiplication by

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- 1 is apparent f rom the fact that J~z, = multiplication by - 1.

Since holomorphic coordinate systems are always C ~*, p -. Jp considered as a (1,1)

tensor on M is C °*. Note that this C ® tensor completely determines the complex

structure on M since a C ® coordinate system ¢ : U -, R ~ is a holomorphic coordi-

nate system if and only if ¢,pJp = JRZ,¢.p for all p ~ U.

DEFLNI~ON. A C OO mapping f : M -- M' from one complex manifold M to

another M' is holomorphic if f * J n = JM'f*.

Lemma 2 shows that this definition is equivalent to the requirement that f be holo-

morphie when expressed (locally) in holomorphie coordinate systems on M and M' .

The definition as given has the advantage of avoiding local coordinate expressions.

Note that if M' is a complex manifold and M a C ® real submanifold then M

has at most one complex structure such that the injection i : M -. M' is holomorphie

since at most one JM can satisfy i , J M = JM'i*, i. being injective. Of course such a

complex structure on M may fail to exist. If one (and necessarily only one) such does

exist, then M is said to be a complex submanifold of M' .

Since the (1,1) tensor J with ,/2 = multiplication by - 1 on a complex manifold

determines the complex structure, it is reasonable to consider such tensors independent

of any complex structure:

D ~ O N . Let M

a C ® (1 ,1) tensor J on M

by - 1 .

be a C °° manifold. An almost complex structure on M is

such that, for every p 6, M, j2 : Mp - Mp is multiplication

L E M M A 3. A real finite dimensional vector space which admits an endomorphism

J : V - V satisfying j2 = multiplication by - 1 is necessarily even dimensional.

Proof. Let X , Y -. g (X ,Y ) be any positive definite inner product on V. Then

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(X,Y) -- H ( X , Y ) defined by H ( X , Y ) = g (X ,Y ) + g(JX,JY) is a positive definite inner

product on V relative to which J acts as an isometry. In particular it follows that any

J-invariant subspace has a J-invariant complementary subspace.

Now note that if X 1 E V and X 1 #: 0 then X 1 and JX 1 are real linearly

independent since if JX 1 = aX1, a E R, then X1 = - j2X1 = - czJX1 = - a2X1 or

ct 2 = -- 1, which is impossible for a ( R. Hence X1 and JX 1 span a 2-dimensional

subspace, which is obviously J-invariant. This subspace has then a J-invariant comple-

ment, say V1, of dimension equal to two less than the dimension of V. The lemma

now follows by an obvious induction on dimension argument.

REMARK. The existence of a J-invariant complement for every J-invariant sub-

space is of course a special case of the complete reducibility of representations of finite

groups.

Lemma 3 shows that any manifold admitting an almost complex structure must be

even dimensional.

L E M M A 4. Let M be a paracompact (even dimensional) C ® manifold with an

almost complex structure JM. In order that J be the almost complex structure on M asso-

ciated to complex structure on M, it is necessary and sufficient that for each point p o f M

there exists a C ® coordinate chart ~ : U -, R 2n, p E U, such that ~*JM = JR2'~ *

everTwhere on U.

Proof.

that if ~ : U 1 - R ~

a n d ~b*J M = JrRZ~dp.

ties

The necessity of the condition is immediate. To prove the sufficiency, note

and d~:U2- . R 2n satisfy O.J~ = JRz, O • everywhere on U1

everywhere on U 2 then O~b-1 : ~b(U1 f3 U2)- . ~(Ulf3Ug_) saris-

( ~ - I ) . j R ~ = ~ . ~ : l y n ~ = ~ . j ~ : l = j R ~ . ~ : ~ = j s ~ ( ~ @ - ~ ) ,

everywhere on @(U1 N U~. By Lemma 1, ~@-~ is holomorphic on the open set

234

dp (UI Iq U2), and Lemma 2 now implies the desired conclusion.

It is possible to fred conditions expressed direc~y in terms of the almost complex

structure tensor J which are necessary and sufficient for J to arise from a complex

structure. These conditions are discussed in the Appendix Z.

2. Basic Examples

Two complex manifolds M and N are biholomorphic if there is a (real) diffeomor-

phism F : M -- N that is holomorphic. It follows from the results in §1 that F-1 is

then also holomorphie. From this, it then follows that the relation of being biholo-

morphic is an equivalence relation. Two complex manifolds that are biholomorphie are

the same object as far as the purposes of complex manifold theory go.

The most fundamental examples for the theory are the following:

(i) C ~. Thisis R z~ with the complex strueture J determined by

n = {(xl,y . . . . :.,y.)}

J( O"~"} a [~yi ) = ayi ' J a Oxia i = l , 2 , . . . , n .

The holomorphie coordinate system (xt + V ~ l y l , . . . , x , + ~/-Zll y,) de t e~ ines a

complex manifold structttre.

(ii) Bn = {(Zl ..... zn) E Cn] ~ ~zj]2 < l}, j= l

the unit ball in C n. Since this is an open subset of C", it inherits a complex manifold

structure automatically. It is important to n~tc that C" and B" arc not biholomorphic

even though they are real diffeomorphic. To see this, note that a holomorphic mapping

F : C n -, B n, F -I- (F1,...,Fn), Fj : C ~ - C is necessarily constant because LiouviUe's

Theorem immediately implies that each Fj ~ constant. (In detail: z -. Fj(a + bz),

a,b ( C, z ( C is constant so Fj is itself constant.)

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(ii S Variations of B". If S is a C ® submanifold that is C 1 close to

{(x>y: .... ,x.,y,,) [ I:x~ + Zy~ = 1}, then the interior Us of S is an open subset of C"

that is real diffeomorphic to B". If n = 1, then Us is biholomorphic to B 1, by the

Riemann Mapping Theorem. But ff n > 2, Us is generically not biholomorphic to B".

The Us so obtained in fact form a coUection of complex manifolds that in a suitable

sense give an infinite dimensional family of biholomorphic equivalence classes.

(iv) P . C , complex projective space of (complex) dimension n. The notation CP"

To define P . C , define an equivalence relation on C "+1 - {(0, .. . . 0)} as is also used.

follows:

(n,...,z,,<) - (wl,...,w,+ O

if and only if 3 k E C such that zj = kwj for each j = 1,2,...,n + I. (Exercise:

Check that this is an equivalence relation.) As a set, P,C is the set of all equivalence

classes of C "+I - {(0,...,0)}I~. Define a topology on P,C by declaring a set

U C P,C to bc open if and only if

{(z~,...,~,+~) ~ c "< - {(0,...,0)} I [(~>..,z,+0] ~ u}

isopenin C "+1, where [ g] denotes the equivalence class of (zl ..... z,+1). (This

is the topology usually called the quotient topology relative to the equivalence rclation.)

P,C is compact becansc the map

{ ( , ~ . . . . . z.<) I~.M== 1}- e .c

obtained by taking equivalence classes is surjective (and continuous by definition).

To define a complex structure on PnC, we exhibit coordinate systems as follows:

For each j , set Uj = {[(z I . . . . . Zn+1)] I zl ~ 0}. Clearly tO Uj = PnC. Define

Fj:Uj-.C n by

[(zl . . . . # . + i ) ] - (z i / z i . . . . . z j z j . . . . . z . / z j ) , e ~: j .

The transition map from Fj(Uj) to Fk(Uk) defined on Fj(Uj A Uk) is easily seen to

236

be holomorphic: it is essentially multiplication by zj/zk. (Exercise: Check details of

this.) Thus the Fj : Uj --, C n maps define a complex manifold structure on PnC.

(v) Submanifolds (of PnC): The definition of (complex) submanifold of a complex

manifold runs parallel with the real theory: A subset N of a complex manifold M is

an (embedded) submanifold if (1) N is a C ® real embedded submanifold of M and

(2) TeN , all p ~ N, is a J invariant subspace of Tr, M (TeN = real tangent space)

of N at p.

This definition condition (2) can be checked to be equivalent to the idea that in a

neighborhood in M of each point q E N, N Cl neighborhood is a slice in some holo-

morphic coordinate system, i.e.,

N n n b h d = { ( z l . . . . . z~) l z j = z l + l . . . . . zn = 0 }

in some (holomorphie) coordinates (zl,...,zn).

A compact submanifold of PnC is called a (compact) algebraic variety. The fact

that a compact complex submanifold of PnC is an algebraic variety in the usual sense

(of being the common zeroes of a set of homogeneous polynomials) is true, but hard to

prove. A converse statement is, however, relatively easy,

If P(zl .... ,Zn+l) is a homogeneous polynomial, then, for ~. ¢ 0,

P(kz 1 .... ,kzn+t) = 0 if and only if p ( z l , . . . , z n + l ) = 0 since

e(Xzl, . . . . Xz.+l) = x a P ( z l . . . . , z . )

where d = degree (of homogeneity) of P. Thus it makes sense to refer to P vanish-

ing (or not) at a point of P,C. If P1 .... ,Pk is a (finite) set of polynomials that are

bomogeneous, set V(P1,.,.,Pk) = the points of P,C at which all the PI,...,Pk van-

ish. The set V(P1,...,Pk) need not be a real submanifold: it could have singularities.

But if it is a real submanlfold, then it is in fact necessarily a complex submanifold.

Rather than considering this fact in full generality, we illustrate with a concrete exam-

pie.

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Set

v = v( ,? + z~ + z~) c v~c

Consider V FI Uj, j ( {1,2,3} where Uj = {[(Zl,Z2,Z3) ] Izj ~ 0} as before, with j = 3,

say. The coordinates on U3 are (zl,zz) with (Zl,Z2)~ [(zl,z2,1)] ( P 2 C . The poly-

nomial vanishes at [(zl,z2,1)] if and only if z~ + z~ + 1 = 0. So V N U3

corresponds in (zl,zz) coordinates to

{(zl,zglz? + z~ + 1 = o}.

This is a complex submanifold in C 2 because, near (wl,wz), w2 ~ O, with

w~ + w~ + 1 = 0, the functions

(z~,z~ + z~ + 1)

form a holomorphie coordinate system and {(zl,z~ I z~ + z~ + 1 = 0} is obviously a

coordinate slice in this coordinate system (what if w 2 = 0? Exercise). So, checking

similarly on V f'l U1 and V f3 U2, we see that V is a complex submanifold of P2C-

It is important to reali:~e that the analogue of Whitney's Embedding Theorem does

not hold in the complex ease. Fast of all, a connected, compact complex submanifold of

C" must be a point. (Proof outline: z 1 IN is holomorphie if N is a complex submani-

fold of C". If N is compact and connected, the Maximum Modulus Principle implies

zj is constant. Think for yourself about why the Maximum Modulus Principle applies in

several variables[) So PnC is not realizable as a submanifold of C a.

You might be inclined to hope that PnC would play some sort of universal embed-

ding theorem role. But this does not work, either. There are compact complex mani-

folds that are not submanifolds (in the complex sense) of any P , C . It is not trivial to

see why, however. See §3.

3. Hermttian and K~hler Metrics

DEFINITION. A C = Riemannian metric g on a complex munifold M is an

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Hermitian metric if for each p E M and each pair of tangent vectors X,Y E MI,, the

tangent space of M at p,

g(X,r) = g(SX,Sr) .

Every complex manifold admire an Hermitian metric. To verify this fact, note that,

since a complex manifold M

Riemannian metric gl. If g

clearly an Hermitian metric.

is paracompact by definition, it necessarily admits a C OO

is defined by g(X,Y) = gl(X,Y) + gl(JX,JY), then g is

The following lemma describes the pointwise structure of Hermitian metrics:

[ .EMMA 1. Let V be a real vector space and J an endomorphism of V satisfying

j2 = multiplication by - 1. Suppose that g is a positive definite inner product on V

satisfying g(X,Y) = g(JX,Pl) for all X ,Y E V. Then

( a ) s C x m ' ) = - s C y x , r ) .

(b) there exists an orthonormal basis for V of the form X 1 , J X 1 , X 2 , f X 2 , . . . , X n , J X n.

Proof. To prove (a), note that g(X,JY) = g(JX,J(JY)) while

g ( J X , y ( . - , , ) ) = g ( J X , - r ) = - g(Jx,r).

To prove (b), one follows the method used to prove Lemma 3 of §1: Let X 1 be a

unit vector in V. Then JX 1 is a unit vector since g(JX1,J'X1) = g(XI,X1). Also

g(XI,JX) = - g(JX1,Xa) by part (a) so g(XI,IX1) = 0. The subspace generated by

X 1 and JX 1 is J-invariant, and hence its orthogonal complement is also J-invariant.

The desired conclusion now follows by induction on the dimension of V.

If g is an Hermitian metric, then the 2-tensor to defined by to(X,Y) = g(JX,Y)

is antisymmetric by part (a) of I.emma 1.

DEFINrYION. If g is an Hermitian metric, the 2-form to defined by

to(X,Y) = g(JX,Y) is the Kiihler form of g.

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L E M M A 2. I f g is an Hermitian metric, then oJ A "'" A ¢o

complex dimension of M) is nowhere zero.

(n factors, n = the

for

Proof. Given p E M, let XI ,JX1 , . . . ,X , , JX " be an orthonormal basis relative to g

Mp. Such a basis exists by Lemma lb). Then

~(x~ , :x~ ) = g(yxi ,yx~) = 1

while

Also

oJ(Xj,JXi) = g(JXj,JXi) = 0 if i # j .

Then

. ( x~ .x j ) = o , ( : x j x j ) = o for a~ i , j .

{~A "'" A~) (X~,ZX~ .... ~:.dX.) = .! ~(X~,rX0 ..... ~(X.wT:.) = ,!

In partiettlar, ( ~ A "'" Ao,) # 0 at p.

PROPOSITION I. A complex manifold is orientable.

Proof. Since any complex manifold admits an Hermitian metric, Lemma 2 shows

that there exists a nowhere vanishing 2n form on any complex manifold of real dimen-

sion 2n.

Actually, it can be shown that a complex manifold is orientable without introducing

any metric concepts. In fact, a holomorphic mapping from a domain in C" to C"

which is a real diffeomorphism onto its image is necessarily orientation preserving on the

underlying R 2n since its (real) Jaeobian determinant is positive (by a calculation with

determinants using the Cauchy-Riemann equations). Thus a covering by coordinate sys-

tems all of whose transition mappings have positive Jaeobian determinant is given

directly by the complex structure.

DEFINITION. An Hermitian metri.c g on a complex manifold is a K i l l e r metric if

240

the Kfi~aler form ~ associated to g is closed, i.e. d~ = O.

The condition d~ = 0 implies a surprisingly close relationship between the metric

g and the complex structure of M. One aspect of this relationship is expressed in the

following proposition.

PROPOSITION 2. Let M be a complex manifold and g an Hermitian metric on M.

Then the following conditions on g are equivalent:

(a) g ~ a KYalder metric

(b) I f D is the Riemannian covariant differentiation associated to g, then DJ = O.

Proof that (b) implies (a): Since Dg = 0 by definition of g and

co(X,Y) = g(JX,Y) for all X , Y, the vanishing of DJ implies that of Dca:

D z ~ ( X , Y ) = Z ¢o(X,Y) - ¢a(DzX,Y ) - ~ ( X ~ z Y )

-- Z g ( J X y ) - g(J(DzX) ,Y) - g(JX,DzY)

= (Dzg) (JX,Y) + g(Dz(JX),Y) + g(JX,DzY ) - g(J(DzX) ,Y) - g(JX,DzY )

= 0 + g((DzJ)X,Y) + g(J(DzX) ,Y) + g(JX,DzY) - g(J(DzX) ,Y) - g(JXJ)zY)

= g(DzJ)X ,Y ) = 0 if D J = O.

Now for any form ~o, the formula

dco= ~ du i A Da/a~ oJ i

holds, where (u i) is a real C ® local coordinate system. In particular, if Dco = 0

then dco = 0.

Proof that (a) implies (b): It suffices to establish the foLlowing formula for an

arbitrary Hermitian metric g and its Kt[hler form w:

1 d ~ ( X , Y , Z ) - 1 g( (DzJ)X,Y) = ~ ~ a,~(JX,JY,Z) .

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For then the vanishing of d¢o implies that g(DzJ)X,Y ) = 0 for all X and Y and

hence that Dzd = O. To establish the formula, note first that, all terms being tensors,

the formula need be verified only in the case that X, Y, and Z are linear combina-

tions with constant coefficients of the standard vector fields associated to a C** local

coordinate system. In fact, thi~ coordinate system may be taken to be the real and ima-

ginary parts of a holomorphic one, say (Xi,Yl,X2,Y2,...,Xn,Yn). Note that if X, Y, and Z

are (locally) such constant coefficient linear annbinations of Max i and O/Oy i then so

are JX, JY, and YZ so that the Lie brackets of any two of the six vector fields X, Y,

Z, JX, JY, JZ all vanish.

Then

g((OzJ)X,r ) = g(Oz(SX ) - d(OzX) , r )

= S(Oz(SX),r) - s(J(OzX),r) = s(Oz(SX),r) + &(OzX,n ' )

= i [(Jx)s(r~z) + z s ( Jx , r ) - rs(JX~Z)] 2

1 + ~ [zg(xjY) + xg(dT~) - ( i r )g(x,z)] ;

and on the other hand

and

a ~ ( x , r . z ) = x ~ ( r ~ z ) + z ~ ( x , r ) - r~ (x~z )

= x s ( n ' ~ ) + z g ( J X , r ) - rg ( Jx~z)

a o , ( z ~ C r . z ) = ( sx )o ( sx~z ) + z o ( s x , o x ) - ( s r ) o ( J x ~ z )

= - ( ~ ) g ( z ; z ) - z s ( x m ' ) + ( ~ ) s ( x ~ z )

The desired formula thus follows.

Proposition 2 implies that the tensor l is invariant under parallel translation rela-

242

rive to a Ka~e r metric; in particular, the value Jp of J at a single point p ( M

together with the Ka3aler metric on M determines J everywhere on M (assuming that

M is connected) and thus the complex structure of M.

If g is a Ka"hler metric, the Ka"hler form ca, being dosed, determines a class in

deRham 2-cohomology. Because taA.--~ (n times) is nowhere vani.~hlng, and hence

a nonvani.~hlng volume form multiple, to cannot be exact. (Detail: If to = dO, then

~o^.--,'~ (n times) = d(0^~o---/v~), n - 1 ~ ' s . So

0 = f d(O^~^---:~) = f ~^--.~ ~ O,

a contradiction.) Thus the cohomology class of m is nonzero. Hence any K/i[hler mani-

fold has nonvani~hlng 2-cohomology in the deRham sense. It can be seen that

M = S~+l x S 2q+l admits a complex structure, all p , q > 1 ( d . [2]). But then M

has zero 2-cohomology and hence does not admit ~ e r metrics (see §10), so such an M

cannot be realized as a submanifold of PnC.

4. Complexlflcation of the Tangent and Cotangent Spaces

DEFINITION. Let V be a real vector space. The complexifu:ation of V, to be

denoted V c , is the complex vector space consisting of all ordered pairs ( v , w ) , v , w E V

with operations defined by

( ~ , ~ ) + (~2,wz) = (~1 + "I,~2 + WE).

(~ + ia) (v,w) = (~v - ~w,~,w + ~ v ) ,

V~

(a)

(b)

~ , ~ E R .

LE .M2~A 1. I f V is a real vector space o f dimension n and v 1 . . . . ,v n is a basis f o r

then

V C has complex dimension n and (vl,0) --- (vn,O) is a (complex) basis f o r V c

V C considered as a real vector space has dimension 2n and (v l ,0 ) , . . . , ( vn ,O) ,

(0,vt) .... ,(0,vn) is a (real) basis for V c.

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Proof. As an example, the complex linear independence of (vl ,0) , . . . , (v, ,0) will be

verified, the other verifications being left to the reader: If for a~, 13e E R, e = 1 .... ,n,

~. (a~ + i l~) (vt,0) = (0,0)

then

or ~ = l a ~ v ~ = 0 and ~,-113~v~ = 0. Hence a e = 0 for all e and Be = 0 for all

e by the real linear independence of the vl .... ,v,.

Since in V c i(w,O) = (O,w) for any w E V, it is reasonable to introduce the

notation v + /~ for the element (v,w) of V c .

DEFINITION. Let M be a complex manifold of complex dimension n, p be a

point of M, and (z l , . . . , z , ) = (xl,yl .. . . ,xn,y,) be a holomorphic cxx~dinate system

defined in a neighborhood of p. Then a/azilp and a/aFilp, i = 1, . . . ,n, are the ele-

ments of M c given by

Ozi

a ~ = 2" + ~ "

When the point p is clear from the context the abbreviated notations a/az~, a/aF~

will be used.

~ c e

a a a = +

axi azi a~

and

244

, : i [ , °} ayl azi aZi

it follows from Lemma 1 that a/azi,...,alaz,, a/a~l,...,a/aZl, being thus a spanning set

of 2n elements, are a complex basis for M c .

Lemma 2. The complex subspace of M c spanned by alazl ..... a/az. is independent

of the choice of holomorphic coordinate system (z1,...,zn). The same is true of the subspace

m ~ d by a/az....,a/oz..

Proof. Let (z'l,...,z'n) = (X'l,y' I . . . . ,x'n,y',) be another holomorphic coordinate

system in a neighborhood of p. Then at p for any e = 1 . . . . . n

a , , = 7 o~., a~., = 7 :~ a + ~ ayj ....... o =1 ax,j axj j=l ax,~ ayj

2 axj + ~ ayj a =~ ay,~ j=~ ay,~ ayj

= - - - i " + i

2 ax,~ ax i 2 ax,e j=1 ay'~ •1 ay'~

j=l [ az'~ ay'e ) axj Ty~

since

axj axj ayj ayj i = + i - -

ax'e ay" ~ ay,j ax, ~

by the Cauchy-Riemann equations. Thus

az,~ j= l ax,~ ay, ~ azj

so a/az't belongs to the complex subspace spanned by a/azl,...,a/az n.

245

ted.

The proof for the subspace spanned by O/OYI,...,O/O/Y~ is similar and will be omit-

If f is a real C O* function defined in a neighborhood of p, then each element of

acts on f by complex linear extemion of the action of the elements of Mp on f

by differention:

O' + i , , )y = (~9 + i ( ~ v , , , ~ M~, .

A second complex linear extension definition gives an action of each element of M c on

complex valued functions defined in a neighborhood of p which are C ® considered as

functions into R2:

(v + /w) (D = (v + / w ) R e f + i(v + ~,)r,,e

= [v(Ree) - w ( ~ ) l + i [w(Ref) + v ( ~ ) l

for v,w E Mp, Ref , Imf as usual. Note that in this sense the last equation in the

proof of Lemma 2 can be rewritten

az, t j .1 zj azj since

1 { °-.~..-i °-..~.-}(xj+iyj) az'~ zj = ~ ax,~ ay, ~

. . . . . . . . + i + i = - i 2 ax,t 8y,t ay,t ax,t ) [ Ox,~ ay,~

the last equality following from again applying the Cauchy-Riemann equations. Similar

computations can be used to show that

a = a ~ a ~

where the ~ in the parenthetical expression denotes the function xj - iyj.

246

I . E M M A 3. Let f be a complex valued function defined in a neighborhood of p in

M which is C ~ as a function into R 2 and (z 1 .... ,zn) be a holomorphic coordinate system

in a neighborhood of p . Then

(a) .~br all e = 1 , . . . , n

a~~ f

(where denotes complex conjugation).

(b) f is holomorphic on the domain of definition of the coordinate system (zl, . . . ,zn) if

and only ~ everywhere on the domain

a aT~ f = 0 for all e = 1 . . . . . n .

Proof. The assertion (a) follows immediately from the definitions. To prove (b),

rccaU that f is holomorphic if and only if f ' J M -- ] R 2 f " everywhere on the domain.

Since

= j 8 a ax~ 8y~ Ox~

is holomorphic if and only if

and

Now

f . 0 _- a (Ref), Omf)

247

f . 0 = 0 ( R e f ) , ~ ( I m f )

$0

JR~/" = -- ~ , ~ (Ref)

JR~. = - ~ , ~ (Ref) .

Thus the necessary and sufficient condition for f to be holomorphic is that

a (Ref) = a 0 a (Ref) a,,~ ~ ( I~() and ~ ('=~') = - oy--7 "

Since

az~ f = 2" ( ~ f ) - ~ (~tf) +

the conclusion now follows.

- ~ O ' f ) + a-t- (Ref)} aye

Here Mt~ is the real cotangent space of M at p.

Since dx i = (dz i + c~.)f2 and dy i = (ar'~i - dz~)f2, it follows from Lemma 1 that

dz i , . . . , d zn , a~"l,...,~"~ form a complex basis for (Mt~)c.

The previous definition is a special case of the following definition:

The process of complexifying the tangent space and relating this complexification to

holomorphic coordinate systems has an analogue in the case of the cotangent space:

DEFINITION. Let (z 1 .... ,zn) be a holomorphic coordinate system defined in a

neighborhood of a point p E M. Then the elements dzx, ~ i E ( M ' p ) c are given by

D d azi = axi + / a y

Def

~ = a x ; - iay.

248

DEFINITION. Let f be a complex-valued function defined in a neighborhood of

p ~ M which is C** considered as a function into R 2. Then df E (M'p) c is given by

DeC ay = a(Ref) + ia(~mf).

L E M M A 4. For any f as in the previous definition, at the point p E M:

i - , l i = I

for any holomorphic coordinate system (zl, . . . . zn) defined in a neighborhood of p.

Proof.

a

azi } o_a_f f = 2" ~xi - i ax,

Similarly

Thus

from which the formula of the lemma follows.

249

LEMMA 5. The subspace of (M'p) c spanned by dzl,...,dz . is independent of the

holomorphic coordinate system defined in a neighborhood of p. The same is true of the sub-

• pace of (M.p) c spanned by ~ . . . . . ~'n.

Proof. Let (z'l,...,z'n), (zl,...,z,) be two holomorphic coordinate systems each

defined in a neighborhood of p ~ M. Then by Lemma 4

j=l j=1

Lemma 3 implies that (a/aFj) z% = (Mazj)~ = o, so that dz' i is a linear combination

of the dzj's. Similarly, azi is a linear combination of azfs.

An element o~ I + i¢~2 E (M'p) c , o~i,o~ 2 E M,p, gives rise to a complex linear func-

tional on (M'p) c by taking

("1 + i,,,z) 6' + i~) = ,,,~(v) - , , ,2(~) + i,~2(v) + i,, ,~(~)

for v,w E Mp. (M-v) c can be thus identified with a subspace of the complex dual of

M c , and by dlmensionality considerations this subspace is in fact the whole of the com-

plex dual of M c . Note that using this identification procedure, dzb...,dzn, d~'l, .... dF n

is the complex basis of (M'P) c which is dual to the basis MSzt,...,a/azn, a/aF1,...,MaY n

of (Mp) c since

dzi(a~j I = 2"1 (dxi+ idyi) [ ''~Oxj ......... + i ayjO }

while

= X ( s i j _ 8ij) + iO= 0 2

= ~ " - i

= 1 ( 8 0 + 80 ) + i(O) = 8q 2

and similarly

while

250

(Here B V= 0 if i S j = 1 if i = j . )

The definition of df E (M*t,)c, where f is complex valued, combined with the

definition of the action of the elements of (M-p) c on M c yields immediately that for

all V E M c

dr(V) = Vf

where Vf is defined as previously. The duality of dzi,...,dzn, a~-l, . . . . a~" n and

O/azl,...,a/sZl .... ,a/azn, 8/8~ 1 .... ,O/Sz n can be interpreted from thi.~ viewpoint also, since

(a/Ozi)~ = O, (a/azi)zj = B V, etc.

5. Complex-Valued Differential Forms

DEFINITION. A complex-valued r-form (or complex r-form) on a real vector space

V is an element of (ArV*) ¢, where ArV * = the real vector space of real r-forms on

V. The (complex) wedge product A : (ArV*) c × (AsV*) c -. (Ar+sV*) c is the complex

linear extension of the real wedge product A : (ArV *) x (AsV *) - A r+~ V*.

The complex wedge product has the associativity and skewcommutativity properties

of the real wedge product.

L E M M A 1. If V is a real vector space of dimension N, then (ArV*) c has com-

plex dimension (the binomial coefficient) [NI; and if {c01,...,toN} is any (complex) basis

for (V*) c then the set of forms

is a basis for

251

{ ~ q ^ " " ^e~, I 1 -< i l < i2 < "'" < i, ~ N}

(A,v.)c.

Proof. let

{O~.~ ̂ - -- A 0i, I 1 ~ ix < "'" < ir < N} is a basis for

( f } . Hence (Arv*)C has complex dimension I f /

¢o i A . . . A ¢oi, , 1 ~ i 1 < "'" < i r ~ ~V, are i N ] U }

ments

0x .... ,0N be a (real) basig for V*. Then the set

ArV *, which thus has dimension

by I .emma 1 of §4. Since the ele-

in number, they necessarily form

a complex basis of (ArV*) c if they generate (A,V*) c . Now, for each i,

O~ = ~ = i aO~j for some (uniquely determined) complex numbers aq, because each

0i ~ (V*) c and the tol's are a basis for (V*) c . Hence

01tA.--A0i, = aiff~o j A---A cqd~ j , =1

from which equation it follows that 0ii/~ "'" A 0g is a linear combination with complex

coefficients of wedge products of ¢oj's. Since any such wedge product is ± a wedge

product of the form ¢ojl A --- A coL, 1 < Jl < "'" < Jr < N , each 0it A ... A 0g belongs

to the complex subspace of (A'V*) c generated by

{,oil A ... A ca], [ 1 ~ Jl < J2 < "'" < J, <- N}. Hence this subspace is in fact all of

(ArV*) c .

If M is a complex manifold of complex dimension n and (z l , . . . , zn) a holo-

merphic coordinate system in a neighborhood of a point D 6 M, then

{c/z 1 .... ,dzn, a~- 1 ..... a~'~} is a basis for (m./,) c . Thus it follows from I_.emma 1 that

{dzit A ... A dzip A ~ A A ... A ~ j , [ I < i 1 < i2"'" < ip ~ n ,

l ~ j i < J2 < "'" < Jq < n , p + q = r}

is a basis for (ArM*p)C.

252

Note: We have to rely on context to distinguish the integer p from the point p E M

in what follows. Twenty-six letters just axe not enough[ Apologies.

LEMMA 2. If (z 1 . . . . . z,) and (w 1 .. . . . wn) are two holomorphic coordinate systems

both defined in a neighborhood of p E M, then dzit A ... A dz~, A c~ji A ... A a~j~ in

(A p+q M'p) c is a linear combination with complex coefficients of forms of the type

dwkl A ... A dw~, A a~ t t A ... A a~ t , (p and q are here f ixed throughout).

Proof. Each dz i at p is a linear combination of dwj's and each dz i a linear

combination of a~j 's (Lemma 5, §4); t!~e result follows directly.

DEFINITION. An element co of (ARM'/,) c is said to be of type (p,q),

p + q = r, ff for some (and hence by L~mma 2 any) holomorphic coordinate system

(zl , . . . ,a,) defined in a neighborhood of p, f l belongs to the subspace of (ArM*p) c

spanned by the set

{dzlA "'" A dz~, A a~jl < ---A azj, I 1 ~ i 1 < i2 < --. < ip < n,

l < j l < J2 < ' ' " < Jq < n}"

It follows from Lemmas 1 and 2 that any element fl of (ArM*p) c can be

expressed as a sum of elements f l0' ,q) , p + q = r , p > 0 , q > 0, of type ( p , q ) ;

moreover, the elements fl~',q) are uniquely determined. Thus well-defined complex

linear maps ~rfp,q) : (ArM*p) C -* (ArM'p)C can be obtained by setting "rrq,,q) = f l (p,q).

Then for any £1, i'~ = Ep+q=r~p~O,q~O Tr(p,q) ~"JL

The elements of (ArM*p) c act on r-tuples vl, . . . . v r of elements of Mp as

complex-valued real multilinear alternating mappings. By complex multilinear exten-

sion, the elements of (ARM*/,) c can be taken to act on r-tuples of elements of M c

now as complex-valued complex multilinear alternating mappings.

A real linear conjugation operation on (ArM'p) ¢ carl be defined as a special case

of a general conjugation on V c, V any real vector space ~ = v - iw, v ,w ~ V.

253

This operation is conjugate linear relative to the complex vector space structure on V c .

An element of V c is an demen t of V C V c if and only if it is invaxiant under the

conjugation operation . Note that if an element of (A~M*t,)c is of type (p,q) then

its conjugate is of type (q,p) since dz i =

ments f~l, D~ of (ArM'p) c .

DEFINITION. A (C ~) complex-value

to : M -* Up¢ M (ArM'p) c such that

(a) to(p) E (ArM'p) C for all p E M

(b) to is C** in the sense that when to

and fit ̂ I~ = ~I ^ ~ for any ele-

r-form to on M is a mapping

is expressed in terms of a

{~,^ "" ̂ ±~ ^~,^--" ^~lI < i~ < --. < ip < ~,

l ~ j l < "'" < j q ~ n , p + q = r}

basis, the coefficients axe C ® (complex-valued) functions of xl,y I .... ,x,,y, on the

domain of holomorphic coordinate system (zl,...,z~). The complex vector space of all

complex-valued r forms on M will bc denoted flr(M). An r-form to E flr(M) is of

type (p,q) if to(p) if of type (p,q) in (ArM'p) c for every p ~ M. The vector sub-

space of fir(M) consisting of all r-forms of type (p,q) will be denoted fiP,q(M).

It is easy to verify that a complex-valued form o~ which has the property that

to~) E ArM*l, for every p E M is a (real) C ® form in the usual sense and that, on

the other hand, any real C °* form to on M can be considered to be a complex-valued

form. Moreover, since dzl, . . . ,dz, , dY 1 .. . . . dY, is dual to O/Ozl ... . . O/az,,

a/OF1 .... ,a/a~-l, the condition b) in ~ e definition of a complex-valued r-form can be

reformulated as the requirement that

. . . . ' p + = "

be a C ~ (complex-valued) function of p E M on the domain of (z! , . . . ,z ,) for any set

254

of indices i1,...,ip, Jl,...dq. It follows that if v 1 .... ,v r are any r complex linear com-

binations of 8/8zI, .... a/Oz.,a/aY1,...,a/aY, with C = coefficients then

~ ( p ) (VlIp ..... vrlt,) is a C ® function of p. In particular,

*,(< < <1 , , . . . , ax n '

is a C ® function. An extended (to arbitrary C O* real coordinate systems) converse of

this last result also holds: ~ : M - Up~ M (A~M*p) c with ~(p) ~ ( A r M * p ) C is C ® if

and only if it is C ® in real C** coordinate expressions.

By complex linearity, the operator d can be extended to fir(M): Explicitly, let

( u> . . . , u z~ ) be a real coordinate system in a neighborhood of p E M and write

(uniquely)

¢0 = f/~ ..... /, dui~ ̂ --- A due , 1 - , : i t < . - - < g ~ : 2 n

where the f ' s

where dfi t ..... /,

d,o e f l "+1 (M)

are C ® complex-valued. Then

d~ = ~, d~ ..... ~ ̂ duq ^ "" ^ du~ I < ~ t < . - - < / ; . : 2 n

is defined as in §4. The usual argument from the real case shows that

thus defined is independent of the choice of the real co~dinate system

(u 1 .. . . . u~,). Also f r o m the real case it follows easily that d thus extended to complex-

valued forms satisfies

d(oJ 1 A ~ = d~lAo~ 2 + ( - 1) roJ 1Ado~ 2

for ,~ ~ I~ ' (M) .

I . E M M A 3.

df f ± i , /' "" ^ dz~, /, 4 1 / ' " A 4 , )

=~"(At'd~iAdzit''"'Ad"~'A~'A'"A~'l~=l az~

255

+ ,=12 ( ( - 1 ) v a-~-dzhA...AdzipAd~iAa~jA...Ad~j,}.O~.

Proof. Since d(dzt) = d(d~t) = 0 for all e = 1,...,n, the formula given follows

by repeated application of d(~ I A ~2) = do~l A co 2 + (-- 1) r ~I A dt~2, ~1, ~2 E IV,

together with Lemma 4 of §4.

Lemma 3 implies immediately that, for ~ ~ D]',q (M),

d~, ( ~+l,q (M) + ~,q+l (M).

DEFINITION. O : D/',q (M) - D V+l,q (M) is by definition

~rp+l, q * d : flP,q (M) - l} p+l,q (M). 0 : Df,q (M) -- ~p,q+l (M) is

"rrp,q+ 1 o d : fg',q (M) -- IlP ,q÷t (M). More generally,

0 : fV (M) - IV +I (M) is the operator ~p+q=r "trp+l,q d ~rp,q and

5 : 1~ r (M) - fV +I (M) is ~,p+qfr "i'rp,q+l d l"tp,q.

In case the meaning is clear, the symbols 0 and ~ will be used without explicit

notation of their domain, as is usually done with the operator d.

LEMMA4. d = 0 + 0 , 0 2 = ~ = 0 5 + d 0 = 0 . If o~ EIIP'q(M) then

~ ~ q ~ ( n ) and 5~ = 0o,.

Proof. All the assertions follow easily from the definitions of 0 and 0, the fact

that d 2 = 0, the direct sum decomposition of IV(M) = ~.p+q=r D]',q (M), and the

behavior of types under conjugation. The details are left to the reader.

Lemma 3 provides a computational description of 0 and 0:

of f dz~, ^ . . - ^ d~. ^ az:, A ... ^ ~,)

= G i=1 Ozi

256

= ~ (- l y a_Cf_ ~:A ... A az~ :,~./~/~ ... A ~ . i=i a~,.

These formtdae can be used to provide computational proof of I_emma 4.

6. Hermitian and Ka'hler Metrics in Complex Notation

The subspace of M c s p . . . e d by a/azllp,...,a/az.lp (where (zI,. . . ,z.) is a holo-

morphic coordinate system in a neighborhood of p E M) is called the holomorphic

tangent space at p. It is a complex vector space of dimension n. By Lemma 2, §4, it is

well defined independently of the choice of (Zl,...,z,) coordinates. Notation: Mg.

If g is a Hermitian metric on M (i.e., a J-invariant Riemannian metric), then g

determines a Hermitian metric on each holomorphic tangent space Mp h in the conven-

tional sense of complex linear algebra as follows: First note that g extends by complex

linearity to be a complex bilinear form ~ on M c . (This form cannot be positive defin-

ite!) Define for V, W E Mv h

g ( v , w ) = ~(v ,w) .

Then g on Mp h so defined is a positive definite Hermitian metric on Mg. (Note: The

use--- for both Mp and Mp h - of the same symbol g can lead to no confusion, espe-

dally since, for real vectors, the conjugation in the definition of g on Mp h would be of

no effect.)

Define, given (z>. . . ,z , ) holomorphic coordinates,

g,y= g Oz i , = ~ az i , Oz.i

Then

1 . ( 0 go-= -~g Ox i - v ' : ~ l aye' ax i +

because

257

: -4 g ~ i ' a-xj + g ~y i ' aYj + ~ g a x i ' ay: - g ay i ' axj

- - T g ° ~ , ' a : +_y__~ a a

• , - = g ~ - x i , g OYi Oyj =g ,j 0

g(v,w) = G(v,w) v,w ~ Mg.

Ass_'x:iated to the tensor G (= ~) is a (real) 2-form

_ v ' ~ .E. g,Te~ A N. 4 j

"No complex tensor could operate on V , W to give g ( V , W ) : there must be a complex

conjugation applied to W first since g ( V , + ~¢/-~1 W ) = - %/-----1 g ( V , W ) !

in the sense that*

and

( g 'aYi ' = g axi- ' = - g ' J = - g a x i ' a Y i

These formulae express g on M h in terms of g on M,,. They also show (when run

the other way) the following: Every Hermitian metric on M~ arises from exactly one

J-invariant (Riemannian) metric on Mp.

Of course, since all these considerations happen at one point p at a time, they axe

really facts about complex linear algebra[

As a tensor, the metric on M~ is given by

i j

258

Exercise. Check that this association is well defined independently of coordinate

choice. This form is just the K/ihler form as defined on page 3.2 (page 2 of §3).

Philosophical remark. This is one of many places where it is possible to get +

signs wrong, or to gain or lose factors of 1/2 or 1/4. Usually it does not matter. But

in cases where it does (later: positive vs. negative curvature), it is safest to check all

computations by simple examples. There are many papers that are supposed to be about

positive bundle (definition later) that are actually about negative ones, etc.

The metric is K~'hler if and only if

0 = dl/~ gffdzi A az: ] ,,.j

log,7 og,7 } ;j,~ a~

Since the terms from the dze have type (2,1) while those from the a~'~ have type (1,2)

each sum

ag, 7 Z ± Aaz, and y. o--ffa A ,A must vanish separately if (and only if) g is a Ka-'hler metric. It follows that: g is a

Kh2tler metric if and only if

ag'7= ag~7 and agiy = agi'~ az~ azi a£e a~j

for all i , j , e . We shall use this soon, in relation to holomorphic normal coordinates.

7. Holomorphic Normal Coordinates

In this section, we want to prove the following computationally useful proposition:

PROPOSITION. Let M be a complex manifold with Hermitian metric g, and let ¢o

be the Kfihler form of g. Let p E M. Then delt, = 0 if and only if 3 a holomorphic

coordinate system (zl,...,zn) defined in a neighborhood of p with

259

a a ] (1) g ~ ' azj

and

= 0 i ~ j

= 1 i = j

oal,,. (2) d azi '

Proof. Conditions [(1) and (2)] imply that dtolp = 0 by the formulae of §6. The

convene is more complicated. We do it as follows:

Choose coordinates (holomorphic) w 1 ..... w, with p -- (0 .... ,0). Changing the w

coordinates by a constant coefficient linear transformation, we can assume condition (1)

to hold. To arrange for condition (2) to hold, we introduce z l , . . . , z , defined by

j ,k

where a~k are complex constants to be chosen later and which are to satisfy a~k = akj.

Note that for each choice of a ' s , the equations define a holomorphic coordinate system

(zl . . . . . Zn) in some (smaller) neighborhood of p: this follows from the holomorphic

implicit function theorem (actually, inverse function theorem). By the (holomorphic)

chain rule:

az i az i aw~ j

and

Hence,

J

go" in z coordinates, hfffor w coordinates,

, ,

260

+ terms quadratic in z's .

At p, g,7 = ho" so condition (1) remains satisfied. For condition (2), we need

agi 7 [ = o .

Again at p (where z's = w's = 0)

( .) ag,7 ah~ 7 - - = + 2 ~ azr azr

because i(a/aw,,a/a~i) = B v at p. Also at p

(**) ag,7 ah,7 ahq- _~ = = + 2 e t j r .

aZr a~r a'Zr

So from (*) and (**) condition (2) holds if

( t ) =Ir =

and

1 ah;7 2 OZr

(tt) - i Or j r = 1 ah,T 2 aE~

If ( t) holds (for all i, j , r) then so does ( t t ) because

ah~7_ Oh17_ Oh iF a~r azr az~

So we need only arrange for (5") to hold for all i, j , r at once. This will be possible if

and only if the righthand side of (5-) is symmetric in i and r. (Remember: We

assumed air was symmetric in i and r, without loss of generality.) In other words,

26t

we must have

a h , 7 _ ahry (ttt)

8zr 8zi

and, if we do have this, then we can solve (t) . But ( t t t ) is true if g

§6). So the proof is complete, m

is K/ihler (end of

While this proof is clear computationally, it is worthwhile to look at the situation

philosophieally, too. In general, Riemann normal coordinates (i.e., exp -1) will not be

real and imaginary part of a holomorphie coordinate system, i.e., J on Mp will not be

taken, in general, to J on M by expp. Of course, since d(expp)lp = identity, J at

p is taken to itself. Also, if the metric is Ka"hler so that DJlt , = O, then [because

Riemann normal coordinates are parallel-at-p (i.e. D(eoord field)It, = 0] the exponen-

tial map is J-preserving at p to one higher order than just the O-order,

dexpp[p = identity. So it makes sense that there is some holomorphic coordinate system

w i t h real and i m a g i n a r y par ts x l ,Y t , . . . ,xn ,y n matching Riemann normal coordinates to

an extra order, i.e., having covariant derivative 0 at p. This is just what condition (2)

of the Proposition gives.

Exerc i se .

it holds that

all at p, all j , e .

.Show that with (Zl . . . . . z,) as in the Proposition and zj = x j + X/'-----f y j ,

8 8 = 0 D a / a x j - - = 0 D°/°xJ ay~ Oy~

a a Do/axl ax---~ = 0 Oa/oy: Ox---~ = 0

8. Basic Examples of K/[hler Manifolds

(1) C n with its standard metric (as R ~ ) . This is easily seen to be KA~ater because

in standard coordinates its metric coefficients are constant.

262

(2) P .C: Denote by P : S ~#+I - C/)" the ~oject ion (to eqmvalence classes) of

s '-'+1 = { (z l . . . . ,=.+1) e c " + 1 Iz l=: l= = 1} onto c t ' " .

(zl,-..,z,,+l) = (x~,yI,...~,,+l,y,,+O,

then (x I ..... Y,+ O and ( - y l , x l , . . . , - y ,+ I,x,+ O together span a J-invariant subspace

S of (C'+l)(xt...y,. 0 = the real tangent space of C n + l = R 2~+2 at (x 1 .... ,Y,+0. The

orthogonal complement of this subspace S projects J-equlvariantly under dP one-to-

one onto the tangent space of CP" at P(zl,.. . ,z,+O. To see this note that the two

have the same real dimension 2n. Also the vector ( - y l , x l . . . . . - y n ÷ l , x , + 0 generates

the (one dimensional) tangent space to the fibre of P through (xl, . . . . Yn+0 because it

is the tangent to the curve

t -. ( c o s t + i s i n t ) • ( q . . . . , z . + O

which is the fibre. So d/'[ orthogonal complement S is injective.

We now define a metric on CP n by declaring d/'{ orthogonal complement of S

to be isometric. It is algebra to check that the resulting metric is C ® Hermitian. To

see that it is K a ~ e r is more tedious if one attempts direct calculation. But the situation

can be simplified as follows:

F'trst note that the unitary group acting on C ~+1 induces an action CP"; and, as

is easy to see, this action is isometric relative to the metric we have defined. It is also

biholomorphic (i.e., each induced mapping is a biholomorphic map of CP" to itself).

Thus we need only check the vanishing of D] (or do ) at one point of C P " - because

the group action is transitive and DJ (or doJ) is an invariant under biholomorphic

isometries. So let us check what happens at [(1,0 .... ,0)]. Writing for convenience

(z 0 . . . . . z,) as coordinates on C "+1 we have the coordinates on CP "

( q . . . . ,z . ) - [ (1 ,z~, . . . ,~ , ) ]

as a holomorphic coordinate system near [(1,0 .... ,0)]. If this coordinate system has

d&~(o ..... 0) = 0, then the formulae in §6 show that DJ = 0 (or dto = 0) at (0,. . . ,0).

263

Now computing the 2ength of O/azi,i > 1, (or the inner product ~(O/azi, a/a~j)) will

involve computing the dP preimage in the orthogonal complement of

x/ r~- -~T; - r .=z~ ' " x/i-z--r--z7

and J of that vector (considered as a real vector). More precisely

(, o( ll g Oz i , az i

where H = the Euclidean metric on C "+I and Q = projection on the orthogonal

complement of {(1/X/~,...,zn/V'-) and J of that vector}. A straightforwaxd calcula-

tion involving order of magnitude argument only* shows that

(°',t g Oz i ' d-z

is constant to second order (i.e., has 0 first derivatives), and similarly for

g a-~i,

(Note: This argument also incidentally shows in detail that the metric on U P n is J-

invariant).

(3) B n and its Poineare metric B" = {(zl , . . . ,z , , ) ]Xlzjl 2 < 1}).

There is a large group of biholomorphie mappings acting on B". Ftrst of all, the

unitary group acts fixing the origin. It is well known (and fairly easy to see) that these

are the only biholomorphic mappings that fix the origan. In addition, the maps (for each

a , a ~ C, lal < 1)

z l - - a (z 1 .... , zn ) - 1 - Ez 1

z 2 ( 1 - la12) ~2

' 1 - ~ z 1 " " '

~.(1 - lal:)~ / 1 5z~ )

*Q(a/azi) = a/az i - term vanishing to 1st order that is perpendicular to

0/vT. . . , z , /V-) ~d J of ~ .

264

are easily checked to be biholomorphic. Combining these two types of maps, we see

that biholomorphie maps act transitively on B n. We define a mettle on B n by setting

the metric at p E B n = a*p go, where go = Euclidean metric at origin and

at, : B" -- B" is a biholomorphic map taking p to the origin. (Any two such differ by a

unitary group dement , so the metric is well defined.)

The metric we have defined is obviously J-invariant. (Exercise: Show it is C=).

To see that it is Ka"hler, it is enough to see that d¢~ = 0 (or DJ = 0) at (0 . . . . . 0)

since all points are biholomorphieally isometrically equivalent to (0 .... ,0). Also, since

the unitary group acts transitively on directions at (0,... ,0), it is enough to check that

DxJ = 0 where X = a/axl[(0 ..... 0). For this, consider the (standard) holomorphic coor-

dinate system (zl . . . . . z,). If

a a O = Ox ~ = D x - - all i , j ayj

then DxJ = 0 since J(a/axi) = a/ayi.

But straightforward calculation shows that g(a/azi, o/aT) is 2nd order constant at

(0 . . . . . O) so D a/axl = D a/ayi = o at (0 .. . . ,0). •

. Curvature Properties of Ka'hler Manifolds

Recall the Riemann curvature 3-tensor

dd R ( X , Y ) Z = (OxO Y - OyO X - O[x,y])Z

and 4-tensor

dec R ( X , r ' ~ , W ) = - S(R(X,Y)Z,W).

On a :'.a"hler manifold, these tensors have special properties rdative to J: Since J is

parallel, it follows that

R(X,Y) (~Z) = JR(X,Y)Z.

265

From this follows

- R(X,YdZ, tW) = g(R(X,Y)SZ,SW) = g(SR(x ,r)z , .m' )

= g ( R ( x , r ) z , w ) = - R ( x , r , z , w ) .

Also since R(X,Y,Z,W) = R(Z,W,X,Y) = R(Z,W,X,Y) on any Riemannian manifold,

we have

R(SX, .n ' ,Z ,W) = R ( Z , W , S X , S r ) = R ( Z , W , X , r ) = R ( X , r , Z , W ) .

So R( , , , ) is a'-invariant in the first two slots and in the last two.

Let X be a unit vector. The number R(X,a'X,X,.IX) is called the holomorphic sec-

t /ona/curvature of the 2-plane spanned by X, dX. If Y is any other unit vector in this

2-plane, thca R(X,a"X,X,£X) = R(Y,./Y,Y,/Y) (Exercise: Check this.) Thus holo-

morphic se~'tional curvature is a well-defined function on the l - invar iant 2-planes in

The holomorphic sectional curvatures at p determine the whole curvature tensor at

p. In fact, a purely algebraic kind of determination holds:

Proposition: Suppose R and T are 4-tensors on a vector space V with

endomorphism J : V -. V 9 ./2 = _ 1. Suppose R and T have the symmetries of a

Riemman curvature tensor (antisymmetric in 1st two, last two; symmetric under

1234 -. 3412; 1st Bianchi) and that both are d invariant in first two and in last two

shots. Suppose also that for all X ~ V

Proof:

Then

R(X,JX,X,J-X) = T(X./X,X,JX) .

R = T .

It suffices to consider the ease T = O. Look at the map

(x , r , v , v ) R(X,Jr,V,SV) +R(X,SU,r,SV) + R(X,SV,r,JU) .

266

V (check this). This is symmetric in X, Y, U and Also it = 0

for X, JX, X, JX etc. By polarization, /~ -- 0. since T = R = 0

Y = V, one gets

(*) 2 n ( x , ~ , x , ~ ) + R(x , . ,x , r , .n ' ) = 0 .

From the Bianchi identity

for X = Y = U = V

With X = U,

R ( x ~ x , r , ~ ) + R ( X , Y , ~ X ) + R ( X ~ ' , ~ , Y ) = 0

SO

R ( x ~ . x , Y ~ ) - R ( x , y . x , Y ) - R ( L ~ . X . ~ ) = O.

Adding minus this to (*) gives

3 R ( x . n ' : ~ , ~ ) + R ( x , r , x , y ) = 0

Replace Y by JY:

3R(x,Y,x,Y) + R ( x j r , x j r ) = 0 .

These last two imply R ( X , Y , X , Y ) = 0. Since sec. curv -- 0 = R -- 0 (in a purely alge-

braic way), it follows that

R = O . •

The proposition we have just proved shows that there could be at most one tensor R

(with all the symme~'ies and J invariances) ~ R(X,JX,X,JX) = + 1 for

X ~ I[X[I = 1. We can in fact exhibit one such. (Note: We shall do so by just writing

one down. But in principle we could compute one, by noting that CP n has constant

holomorphic sectional curvature, since the biholomorphic isometrics act transitively on

the J-invariant 2-planes. So we could just compute the curvature tensor of CP n, and

even at one point, in fact. Then we would have at most to multiply it by a constant).

With g as the real J-invariant inner product set

Ro(X,Y,U,V ) = ¼~(X,U)g(r,v) - g(X,V)g(Y,U)

+ g(X,JU)g(Y,r¢) - g(X,rC)g(Y,JV)

267

+ 2g(xa-v)g(v,r¢)).

A tedious but easy check shows R 0 has the required symmetries and satisfies

Also)

R o ( x , . r x , x , . t x ) = g ( x , x ) 2 .

R0(x ,Y ,x , r ) = . { s ( x , x ) g ( Y , Y ) - g ( x , Y ) z + 3 g ( x , ~ ' ) 2 } .

This last formula has a pleasant geometric meaning.

Suppose P is a 2-plane. Then we define the angle ap between P and IP by

ap = minim~n angle between X ( P and Z E JP. Then

(~¢a)) = [ s ( x ~ ) l

where X, Y is an orthonormal basis for P. (Exercise: Prove this.) Thus the sectional

curvature (for R0 as curvature tensor)

K(P) = 1 ( 1 + 3 cos2(otp)) .

We can summarize all this as follows: If at a point p ( M, M a Khmer manifold,

all J-invariant 2-planes have the same sectional curvature c then the curvature tensor

at p is cR o and for any 2-plane P in Mp

tc(t') = (u4) (1 + 3 cos 2 ( ~ t ) ) c .

The next result shows that a Kb2fler ~anifold of C-dimension - 2 can satisfy the

previous condition at each point only in a special way:

PROPOSITION. I f M is a (connected) K&hler manifold with

dimRM = 2 dimc2M ~ 4 and if, at each point p E M, R = cp R o for some constant

c, then in fact c is independent of p!

268

Proof. Let Ric denote the Ricci tensor of M. Compute for an arbitrary

x , x ~ M,, Ilxtl = 1

n

R i 4 X , X ) = a ( x . z x ~ ) + E [ R ( X , r j ~ , r j ) + R ( X , ~ j . X ~ j ) ] j=2

where X , ,IX, I" 2, J'Ir 2,...,Yn, J]', is an orthonorlnal basis of Mp. Now

R ( X , J X , X , J X ) = c v while

R ( X , r j , x , r j ) = a ( x , n ' : , x , ~ j ) = t / 4 cp .

So

f n + l ) R~<x,x) = ~ . + 2 [ ( . - ~)/41c. = I - T - i t . .

In particular, Ric(X,X) is independent of X ~ Mp, Ilxll = ~. so M is F.~-,tem (at

p). By a classical theorem the quotient % = {U(n + 1/2)} [Ric/g] is independent of p.

Of course, none of this discussion about independence of point has any relevance in

case n = 1 (a 2-climensional manifold in the sense of real dimension).

Our three basic Ka"hler maniiolds - C n, CP n, B n - all have constant holo-

morphie sectional curvature. C n is of course flat: R ---- O. C P n has comtant holo-

morphic sectional curvature (because the biholomorphic isometrics act transitively on the

J-invariant 2-planes). CP n is simply connected and compact so it cannot admit a

(necessarily complete) Riemannian metric with sectional curvature ~ 0. Hence its holo-

morphic sectional curvature (which is constant) is > 0. Similarly, the complete metric

on B" cannot have positive curvature, since it would be positive bounded away from

zero, contradicting the noncompactness of B n. SO Rs~ = cR o, c ~ O. It cannot be the

case that c = 0 because then B" would be biholomorphicaUy isometric to Cn:

namely, the exponential map would be such a biholomorphic isometry. So c < 0 in

this case.

269

These three examples in fact are all the possibilities for constant holomorphic sec-

tional curvature, up to coverings and constant factors. Specifically, the following result

holds:

PROPOSITION. Any two complete n-dimensional C simply connected Ka-hler mani-

folds with the same constant positive holomarphic sectional curvature are biholomorphically

isometric to each other.

Proof. Let M 1 and M 2 be two such manifolds. Choose Pl ~ Mi and P2 E M 2

and let T : Mpt - Mp2 be an isometric linear transformation that commutes with J.

(Such a transformation can be obtained by choosing orthonormal bases

XI, JXI,...,Xn, J'X, and YI, JYI,---,Y,, JY, and defining T(Xj) = Y], T(JXj) = JYj

for all j = 1,...,n.) The transformation T also takes R m to Rp2, by our previous

results. Moreover, since JM: and JM: are both parallel and since the curvature of MI

(and M2) is determined by the metric, I and the equal constant holomorphic sectional

curvatures, it follows that both MI and M 2 have parallel curvature tensors. By simple

connectivity of MI, and standard considerations*, T must be the differential of a

locally isometric sttrjective covering map T : M1 - M2. T must be holomorphic (Exer-

cise: Prove this by using parallelism of I to show dT o JMI = JM2 ° dT at each point of

MI, since the equation holds at p). Since M2 is simply connected, T must be injec-

tive. "

10. K~hler Submanifolds

Let N be a complex submanifold of a complex manifold M. Since by definition

JNIq = the restriction to Nq of JMIq (Nq being /-invariant), it follows that the res-

triction to N of a Herm~tian metric on M is a Hermitian metric on N. Also, the

Kghler form of the metric on N is the restriction to N of the Kz-'hler form of the

"of Riemannian geome~y.

270

Hermitian metric on M. Since d N = d M~V, it follows that if the metric on M is K~ihler

then so is the metric on N. The relationship between the curvature of the metric on M

and that of N is much closer than in the Riemannian case, where, in the case of large

codimension, at least, there is no relationship. The relationship in the K~'hler situation

comes from special properties of the second fundamental form.

Let S ( X , Y ) = Dx M Y - Dx N Y be the second fundamental of N in M. Then

S (X , JY ) = DMx(JY) - D~(.IY) = JM DMx r - JN DNx Y

By symmetry,

= Ju(Ox M r - Ox N Y) = J s ( x , r ) .

s ( z x , r ) = s ( r ~ x ) = J s ( r ~ ) = ~ s ( x , r ) .

For any submanifold, the se~ctional curvatures satisfy ( X , Y orthonormal, P = span

of X,Y) :

Ks(e) = RM(x,r~,Y) + g M ( S ( X ~ ) , S ( r , r ) ) -- s M ( s ( x , r ) , s ( x , r ) )

= KM(P) + s u ( S ( X ~ ) , s ( y , y ) ) - g u ( S ( X , r ) , s ( x , r ) ) .

In our ease, if Y = ./X then

I~N(P ) = K ~ ( P ) + gM(S(X,X) ,S(JX~rX)) - g M ( S ( X ~ W ) , S ( X ~ ) )

because

= K ~ ( P ) - 2 g ( s ( x , x ) , s ( x , x ) )

and

s ( J x , z x ) = - s(x,x)

g M ( S ( X , f X ) , S ( X , J X ) ) = g M ( J S ( X , X ) , J S ( X , X ) ) = g M ( S ( X , X ) , S ( X , X ) ) •

In particular, the holomorphic sectional curvature of N is always ~ the corresponding

sectional curvature of M. Let us compute the trace of S: we can of course compute

relative to any basis so we use a basis of the form XI,JT~I, . . .Xm,fXm, rn = d i m c N .

Then

271

T r a c e s = s ( x z , x o + s ( s x ~ , : x ~ ) + ... + S ( X . , X . ) + s ( s x . , s x . ) = o

because S(,/Xl,JX1) -- - S(Xl,Xl). So N is a minimal submanlfold of M. (In case N

is compact and so is M, it can be shown that N is absolutely a r e a minimizing in the

homology class it represents in M, this homology class being necessarily nontrivial if N

is not zero dimensional.)

l l . Holomorphie Vector Bunfll~ and Hermitian Metrics and Connections

A holomorphic vector bundle is defined just as is a topological vector bundle with

fibres C-vector spaces with an additional restriction, that the transition functions be

, i t

holomorphic functions of the point in the mznlfold. More explicitly, with B -. M, M a

complex manifold bundle, we suppese given a trivfli~ng cover Ux and maps:

ex : =-1(Ux) - Ux x C k

with ~I ° ¢x = ~/~-1(u~) when ~r I = fn~st factor projection and the linear maps,

defined for p ~ Ux~ N Ux~ of C k - C k by ~z ° ¢x2 ° (¢~I I P x C ~) to depend holo-

morphicalIy on p. (Notes: "rr 2 = projection on the second factor; since linear maps

C k -, C k are uniquely associated to k × k C-valued matrices, it makes sense to speak

of such maps being holomorphic: it just means that each matrix element is a holo-

morphic function.)

Notation: "tr 2 o cbx2 ° ( ~ 1 I P × C ~) = fx~x~ E C : .

A Hermitian metric on a holomorphic vector bundle B -. M is a C °O family of Her-

mifian metric (in the standard linear algebra sense) on each fibre ~r-l(p), p E M.

A R-vector bundle with Riemannian metric in general admits a wide variety of

metric-preserving connections, i.e., connections for which parallel translation preserves

inner product. (Recall: The unique Riemannian connection on TM, M a Riemannian

272

manifold is made unique only by imposing the additional condition of torsion 0.) Simi-

larly, a complex Hermitian vector bundle admits many Hermitian-mctric preserving con-

ncctions. In the case of a holomorphic Hcrmitian vector bundle, however, there is a

natural way to select a unique metric-preserving connection from among the many possi-

ble metric-preserving connections.

~r

DEFINITION. A connection on a holomorphic vector bundle B -. M is type (1,0)

if its connection forms relative to a local holomorphic f rame in B are type (1,0).

It is easy to cheek that the definition does not depend on which holomorphic local

frame is used.

The basic uniqueness and existence result is the, following:

THEOREM. If B -. M

(fibre) metric on B, then 3

ing.

is a holomorphic vector bundle and if h is a Hermitian

a unique type (1,0) connection on B that is metric preserv-

Proof. Choose a local holomorphic frame, i.e., a trivialization

4, : "rr-l(U) - U x C k and set crj = ~b-l((0 .-- 1 "" 0) so that crl,...,0r k are holo-

morphic and span "n'-l(p) at each p E U. Set

h a l 3 = h ( c r a , o r B ) 1 ~ ct - - k , 1 < 13 ---- k .

Define the connection forms of a covariant differentiation (connection) D to be those

determined by

(i.e.

k Do'a = ~ co~o'S, ct = l,...,k.

8=I

k = y. (x) "8)

8=I

273

Then D is metric-preserving if and only if

dh a = +

o r

k k

"y=l &=l

(Note that the ~ is conjugate in the second sum because h is conjugate linear in the

second variable.)

If ~ ' s are type (1,0) so that ~ 's are type (0,1) then we must have

k

.,1=,1 and

k

~=1

If the fast set of these equations hold, then so do the second because

k k

&=l ~=I

On the other hand, the fact that the matrix (h.~¢) is invertible (being positive definite)

means that we can choose the to~ in one and only one way so as to make the fas t

equations (5") work. So we have local existence, and uniqueness for the desired connec-

tion. Global existence (and uniqueness) follows as usual. •

In a holomorphic frame, we have from th8 previous

a

274

where h ¢a is the inverse of h (so ~ x hpx h x~ = 8~).

An interesting special case is that of holomorphic llne btmdles. (Note: Line bundles

in the C sense are more interesting than those in the R-sense. The latter have discrete

(± 1) structure group, i.e., are reducible to that group. But in general C bundles are

reducible only as far as S 1 C C*, not to locally constant transition functions.)

In the line bundle case, h is a 1 x 1 matrix. Also

to] = Uhr~ ah n = a(log hn)

(for log as in the real sense: hn > 0). In a different frame, hll changes to ffhn, f

a (nonvani~hing) holomorphic function. It follows that

aa(log hll f ~ = aa(log hll + log f + log ~ = aa log hll ,

where log f and log f can be any fixed local branch of (holomorphic) "log". So

a(log

is in fact a globally deemed type (1,1) form.

Associated algebraically to the well-def'med form a01og htl is the Hermifian form

] (tt) - i j=1 ~" az~a~ log h n dziL~azj.

We shall c.all this form the Hermitian curvature form. The association is essentially that

of metric to K/itfler form. But to avoid algebraic detail, it is more convenient simply to

compute directly that the Hermitian form is independent of local trivialization. This is

easy foLlowing the line of reasoning used to show that aalog hll is well defined, and it

is left to the reader. (This approach also avoids the possibility of sign errors, which

have plagued the transition from type (1,1) to Hermitian forms in the literature.)

Special interest is attached to Hermitian holomorphic line bunches for which the

Hermitian curvature form indicated is definite, either positive everywhere or negative

everywhere. We make a formal definition:

275

DEFINITION: A holomorphic Line bundle B is positive (respectively, nonnegative)

if for some Hermitian metric on B the Hermitian form (¢¢) is positive definite (respec-

tively, nonnegative definite). The bundle B is negative (respectively, nonpositive) if for

some Hermitian metric on B the Hermitian curvature form is everywhere negative

definite (respectively, nonpositive definite).

The logic of the negative sign in the definition of the Hermitian curvature form is as

follows. Conventionally, it has been the practice to regard bundles with global holo-

morphie sections positively, having sections being a good property. Now if a Hermitian

line bundle over a compact complex manifold has a nontrivial holomorphie section s

then for a local frame field tr 1 we can write s=ftrld holomorphie, and then, where

s#O:

1.2 1 2 az,a~ log hal a~,Q~j= ~ az,a~ i j = l id=1

log(h f ]

is nonnegative definite.

where the Hermitian form

{a2 ] 2 aziaEJ log hll d z i ~ ij=l

must be nonpositive definite or equivalently points at which

az,a j log hll ± , 0 4

Note that no such reasoning occurs at the minimum of I[sl[ 2 in general because the

(Here [HI 2 is the Hermitian norm squared of the section s.) The last-written form

must be nonpositive definite at the point(s) where the global function Ilsll 2 attains its

maximum. (Exercise in calculus: Prove this.) In particular, there must be points of M

276

minimum may well be zero, in which case log [[s[[ 2 is not defined at the minimum.

What we do obtain by following the above pattern is that if s is a holomorphic section

which nowhere vanishes (so that minimum Ilsll:>0 and log Ilsli 2 is def'med and C ®

globally) then the Hermitian curvature form must be nonpositive definite somewhere.

Of course, such a section exists if and only if B is the trivial line bundle. Thus we see

that a line bundle on a compact complex manifold that is positive or negative cannot be

trivial.

Every complex manifold M has a naturally arising holomorphic line bundle on it.

This is the bundle of forms of type (n,0); this bundle is called the canonical bundle of

M and denoted by K or, when the manifold needs specification, KM.

If M has a Hermitian metric g then KM can be given an associated Hermitian

metric as follows: Let (zl,...,z~) be a local coordinate system on M and set

g,T=z(a/az,a/a~) as usual. Then put

Ile, = Udet (g ,7 ) ,

where det(g,7)= the determinant of the matrix (g~T),l<ij~n. A slightly tedious but

routine calculation shows that this detrmition yields a well defined C ® Hermitian metric

on KM. We shall car this the canonical metric on KM (associated to the metric g on

M).

The Hermitian curvature form of the canonical metric on KM is closely related to

the Riemannian curvature tensor for a Ka"hler manifold M. To make this relationship

explicit, let R be the Riemannian curvature (4-) tensor of the Ka-hler manifold M.

Define, as usual, the Ricci tensor Ric of M by (at each point of M)

2n gic(X,I") = R(X,e i ,Y,e i ) ,

i = l

where {eili = 1 .... 2n} is an orthonormal (real) basis for the real tangent space Mp of

M at p and X,Y~Mp. It is easy to check that Ric(X,Y) is independent of the choice

277

of orthonormal basis. It is also easy to see, using the d.symmetries of ~ e curvature ten-

sor, that Ric is an Hermitian form, i.e.,

The Hermitian real form Ric extends by complex linearity to Mp@C. On the holo-

morphic tangent space of M

process:

at p we define a C-Hermitian form by a now familiar

I

(z,w)-Ric(Z,W).

The promised relationship between the Hermitian curvature of K M and the curvature of

M is given now by the formula (with G=det(g~y), l< id~n) :

L i I az,a (1ogG) az,®, jj ~ ~C ~Z ~W~

for all Z,W in the holomorphic tangent space of M. Here the left hand side is exactly

the Hermitian curvature form of K M evaluated on Z,W because the metric of K M is

(locally) I/G and log( I /G)=- logG. Thus, for example, if M has positive sectional

curvature and hence positive definite (real) Ricci form, it follows that K M is a negative

bundle. Specific examples of this: pn C, all n. The formula just displayed caa be pro-

ven by a dedious but straightforward calculation, using the standard rules for differen-

tiation of determinants.

Note: In case M has real dimension 2, then the formula reduces m the familiar

expression for the Gauss curvature for a metric of the form G(dx2+dy2), namely

k = - 1/2A (log G).

We can see that the signs check out correctly in our relationship between Hermitian

curvature of the canonical bundle and the Ricci curvature of the manifold by considering

complex projective space. It has positive sectional curvature and hence positive Ricci

curvature, in its usual Fubini-Study K/ihler metric. Thus, according to our calculation,

its canonical bundle K is negative (in its canonical metric). Negative bundles cannot

278

have nontrivial holomorphic sections: and sure enough there axe no nontrivial holo-

morphic (n,O) forms on complex projective n-space P,C.

Looking at this same example the other way around, let us consider K~, the dual

bundle of KM. We can put on K~ the dual of the canonical metric, namely, if o- 1 is a

local nonvanishing section of K~ and cr 2 is a local nonvani~hing section of /C M we set

II iII 2 = I i( 91/II,,:II •

It is easy to see that the Hermitian curvature of K~ in this metric is just - 1 x the

Hermitian curvature of KM. (This hold thing works for any line bundle and its dual:

exercise.) Thus g ~ is a positive bundle. This is as it should be since K~ has non-

trivial global holomorphic sections (obtained by wedging together holomorphic vector

fields). Exercise: Figure out one such section on PtC = the Riemann sphere.

We have already noted that line bundles with a nontrivial holomorphic section must

have positive Hermitian curvature form somewhere (in any Hermitian metric). There is

a profound and important converse of this, first established by K. Kodalra.

By definition, the tensor product B1 @ B2 of two line bundles is the line bundle

whose transition functkms are the products of the transistion functions of B 1 and B 2.

(Exercise: Q on bundles corresponds to addition of Hermitian curvature forms. For-

mulate this precisely and prove it.) Along these same lines, it can be shown that suffi-

ciently high ~-powers actually give an embedding into projective space in the following

sense: Let B-M be a line bundle over a compact complex manifold M. Let H°(B)=

the C-vector space of holomorphic sections of B. This is actually a f'mite-dimensional

vector space. (It is easy to see, for instance, that its unit ball is compact m the norm

ll, IL =d" ll, il:) where II, II is the point se norm in an arbitrary Hcrmifian metric

on B and the integral over M is relative to the volume form of an arbitrary Hermitian

metric on M).

279

Choose a basis s b . . . , s k for H~(B) and suppose (additionally! - this doesn't have

to happen) that there is no point x EM with all sj(x)=O, j = l . . . . . k. Then we can

define a map £e'.M-.Pk_IC as follows. For xEM, choose a jE{1 .... ,n} with s j (x)#O.

Then set the image SB(x) of x= the point with homogeneous coordinates

s l (x) /s j (x) , . . . ,1 . . . . ,sk(x)/s/(x), where the 1 is in the j th slot. That is, the image of x

is the image of (sl(x)/sj(x) . . . . . 1, . . . ,sk(x)/sj(x)) eC~-{(0 .... 0}, under the usual quo-

tient map C~-{(0,... ,0)}--Pi_IC. The map £e is holomorphic. A different choice of

the basis sj changes £s only by a linear isomorphism PnC.

Then the following important theorem holds, the famous Kodaira Embedding

Theorem:

If M is a compact complex manifold and if B-.M is a positive line bundle, then

there is a positive integer t such that 8st is defined and is a holomorphic embedding

of M into a complex projective space.

As noted, PnC has a positive holomorphic line bundle on it, namely K ' . If M is

a compact complex submanifold of PnC then it is easy to see that K" IM is positive.

(In fact, the restriction of a positive bundle is positive: prove this as an exercise.) Thus

we see from the Kodaira Embedding Theorem that a compact complex manifold is biho-

lomorphic to a submanifold of some complex manifold if and only if M has on it a posi-

tive holomorphic line bundle.

The proof of the Kodaira Theorem is highly nontrivial and will not be given here.

In case that the canonical bundle is positive or negative or a trivial bundle, it is

natural to ask whether it is possible to make the Hermitian ca,rva~xe form of the canon-

ical bundle equal to a scalar multiple of the metric itself. In particular, one could ask

whether there is a Kh'hier metric g on M such that, for some c E R,

Ric (z,w)--- cg(z,w)

for all (complex) vectors W E M e ~ C, all p 6 M. In this set up, c < 0 corresponds

280

to K M positive, c = 0 to K M trivial, and c > 0 to K M negative. This is known

not to be possible in general if K~ is negative. But if Kv/ is trivial or,~pcysitive then in

fact such Ka"hler metrics always exist. This important and deep result was shown by S.

T Yau after an earlier conjecture by E. Calabi. (The case c < 0 was established

independently of Yau's work by T. Aubin.) These results are discussed in detail in refer-

ence [1].

Another particulary interesting holomorphic vector bundle is the holomorphic tangent

bundle of a complex manifold M, i.e., the bundle with fibre at p = by def'mition to

the holomorphic tangent space at p = by definition the C-linear span of

O/Oz I . . . . . a/Ozn, ( z l , . . . , zn) a holomorphic coordinate system near p. Suppose M is

given a Hermitian metric g, i.e., a J-invariant Riemannian metric. Then T h M, The

holomorphie tangent bundle of M, becomes a Hermitian holomorphic vector bundle,

with the Hermitian metric determined by g (as in an earlier section). The Ka"hter case

of this situation is especially nice, as we shall see after we prove the following lemma.

"It

L E M M A . Suppose B -. M is a Hermitian holomorphic vector bundle and EI , . . . ,E k

is a basis f o r the f ibre Bp at a point p ~ M. Then 3 a local holomorphic f rame

o" 1 . . . . ,ar k

(1)

and

(2)

Proof.

¢rjlp= Ej, all j = 1, . . . . k

D~ylp = 0, all j = 1 . . . . . k .

Choose a local holomorphic frame "h .... ,~.k with

all j = 1 . . . . . k. Now compute

o ( E = E + E F P P ~,~

This = 0 at p if and only if

281

d3 m o = - the ~8 component of ~'~ fa0 o)~ -~

= _ E / ~ o , , g .

P

Since thi~ last form is type (I,0) we can choose a set of holomorphic fal3 with

/~% = and a ~ to sat~fy me equ~tiom just given. Then D(E~/~e "~)Ip = 0 aU

ot = 1 . . . . . k. Since f is then invertible near p,

cr~ = ~ ~13 'tO 0

fits the requirements, u

If M is a Hermit Zan a manifold (i.e. M has a J-invariant Riemannian metric

attached) then the Riemannlan connection on M induces by complex linear extension a

connection on TM @ C, the complexified tangent bundle of M. If M is Kz-'hler, then

ThM is a parallel subbundle of TM @ C. (Here a subbundle is parallel by definition if

parallel translation of a vector in the subbundle remains in the subbundle.) This follows

because J is parallel so its eigenspaees are preserved by parallel translation.

Moreover, again if M is K~ihler, the Riemannlan connection on ThM determined

by considering ThM a s a Hermitian vector bundle and finding its unique t3q~e (1,0)

connection. The proof of this fact is easy: Choose a holomorphic normal coordinate

system at a point p, say (zl, . . . . zn). Then for the Riemannian connection D

by definition. Let V = the Hermitian con~ection for TriM. Then also

V 0 = 0 j = 1 , . . . , n . 0z i

~ U S e

dg azi '

282

So D = V .

The convene of this is also true. For this, let M be a Hermitian manifold - with

Riemannian metric (d-invariant) = g and Riemannian metric (d-invariant) = g and

R.iemannian comlection D. Then D induces a connection /5 on TaM by setting

/saW = the projection of DzW on TaM ,

where projection is relative to the canonical splitting TMp ~ C = TaMp @ TaMp. (This

also happem to be orthogonal projection relative to g extended to TM ~ C as a Her-

def mitian metric.) In this setting, we can see that if /5 = V ( V = the Hermitian con-

nection) then M is K/ikler. [Note: On the previous pages, we showed that if M is

Ka-hler, then /5 = D = V.]

To prove this, choose a local holomorphic frame crj in TaM which is orthonormal

at p and has V = 0 at p (by the I.emma). Then, since V = / 5 by hypothesis, we

have

But /5 is a length compatible connection [i.e. g(/sxZ,W) + g(Z,15xW) = Xg(Z,~/)]

became D is and the projection involved in /5 is orthogonal. So

dg(crj,~'k)lp = 0 j ,k = 1 .... ,n .

Moreover, /5 has torsion 0 because D does and Lie brackets of vector fields

preserves types , i.e., [Z,W] E TaM if Z ,W E TaM. Since /scrj[t, = 0, it follows that

[aj,cr~:] = 0 at p, all j , k. Now given a holomorphic local frame crj with

[crj,crk]lp = 0, there is a holomorpl~c coordinate system (z!,...,z,,) such that

crjlr, = ,9/azjl p and moreover o'j - O/Ozj vanishes to order two at p. Then (zj) will

be (in our case) a hotomorphic normal coordinate system at p. So M is Khlaler. It

Appendix: Integrable and Nonintegrable Almost Complex Structures

283

For convenience, we based our decomposition of differential forms into types (p,q)

on the use of complex (holomorphie) coordinates (zt, . . . ,z,): i.e. {(p,q) forms} = span

of dzil A "'" A dzi, A azj~ A "'" A a~j¢ However, it is important to realize that this was

only for convenience. The whole question of types of forms is after all a strictly point-

wise item, and one should expect to be able to consider it algebraically. Specifically, we

can consider it as follows:

Let V bc a real 2n-dimensional vector space with an endomorphism

J : V -. V 3 j2 = _ 1. Then I/* G C = direct sum of two subspaces determined as

follows: Let XIJXI,...Xn,YXn be a basis for V and (oI,¢o 2 --. ~-i,¢o~n be the dual

basis. [So if Xl = a/Oxl, . . . . o h = d x 1, t~ 2 = d y 1 etc.]. Then

V* G' C = span{toz~_i + X/L-11ca2k, k = 1,...,n}

e sPan{0,2k-1 - V'2-ilo~2k, k = 1 . . . . . n ) .

This d ~ p o s i t i o n is easily checked to be invariant under changes of the

XI, JX 1 ... . ,X,, .Of, basis (to another of the same form). This invariance can be in fact

checked without computation as follows: The span of X 1 + X ' / - '~ J X 1 . . . . ,X n + %/_-Z'~ Xn

which corresponds to a/a~'s is the - V'-L-~ eigempace of I extended by complex

linearity to V* G C: e.g.,

= - q - - - i (x~ + q ~ - i J x a ) .

Similarly

J ( x l - x/---y JX!) = - x / - - ~ j2 x l ÷ J-X1 = v'L--fl x~ + JX~

So

= V ~ t (xl - V - 1 JX)

V G C = ( -V ' -~ I ) eigenspace of J ~ "v/-L--11 eigenspace of J

284

The decomposition of V* @~ C is then determined by

det v I'° = span{~z~-i + V-- '~ ~z~, k = 1,_.,n}

[e.g.

={,~ ( v * ® c l~,(x) = 0, an x ~

- %/=~1 eigenspace of J on V ® C}

a~((a/a~) + x/--T (a/aye)) = o1.

def v °'1 = s p a n { ~ z ~ - i + X / - ~ l ~ 2 , , k = 1 , . . . , n}

= { ~ ( v * ® c I ~ ( x ) = 0 , an x ,

+ X/---'~ eigenspace of J on V Q C } .

[e.g. ,~((a/a:q) - q-C~l (a/aye)) = 01.

Thus V* @ = V 1,° @ V e,1 and this is basis-choice independent.

This decomposition of V* Q C gives rise to a decomposition of the whole complex

exterior algebra (over C) Ac(V* Q C) and hence to a type (p,q) decomposition as

before.

Now suppose M is a C ® manifold on which there is a C ® family of endomor-

phisms Jp : Mp -- My, p E M, 3. Ip 2 = - 1 for all p E M. Such a family is called an

almost complex structure on M. (Note: Not every manifold admits an almost complex

structure.) We specifically now do not assume that the almost complex structure J comes

from a complex structure on M, i.e., a covering of M by coordinate charts with holo-

morphic overlaps. The question then naturally arises, when does such a general almost

complex structure in fact come from a complex structure? Necessary conditions are not

hard to f'md:

Let Y~',q = the set of C ® C-valued differential forms co on M ~ oJlt, is type

(p,q) at p for all p E M. Then 3 r°,° = C-valued functions on M, etc. Note that

285

70,0, 71,0 and 7 °,1 generate (under A)locally Y= Up,q 5 rt',q. Also

d70,o c y~,o® 70,I = all i -forms

d71'0 C ~r1,1 ~ ~2,0 ¢ 70,2 = aJ~ 2 - f o r m s

eS ° : c : : ® : , 0 ® 70,2

It follows (from the local generation of D that

d~ t'p'q C )~p+l,q O ~p,q+l @ ~rp+2,q-1 @ jrp-l,q+2 .

For a complex structure

d.T p'q C 5 rp+l'q @ ~-p,q+l .

So a necessary condition for an almost complex structure J to come from a complex

structure is that

(*) dF' ,q C F '+l,q q~ ~,,q+l

when 7 is decomposed according to J types.

(Note: If dim R M = 2, then 5 rp,# = 0 if p ~ 2 or q > 2. Hence the necessary

condition (*) is automatically satisfied.)

It is a fact - - not emily proved, however - - that the necessary condition (*) is in

fact sufficient. An almost complex structure satisfying (*) is said to be integrable. Then

we have Theorem (Newlander-Nirenberg): If J is a Coo integrable almost complex

structure on a C oo manifold M, then there is a complex structure on M such that the

J-mapping associated to this complex structure = the given almost complex structure

J .

It is of interest to look at what this theorem says in the dimRM = 2 case: J

determines an orientation on M. Moreover, we can construct a J-invariant metric

(Riemannian) on M by setting

g(x ,Y) = G ( x , r ) + G(s-x,sr)

286

where G is an arbitrary Ricmannian metric on M. Now if z = x + /y is a J-

holomorphic coordinate system on M i.e. if J(a/ax) ~- a/ay then the metric g is

given by

~.2(d, x2 + dy 2)

where ~ is a C ® function. The converse also holds: If (x,y) is an oriented real coor-

dinate system and if g(a/ax,o/ax) -- g(a/ay ,a/ay) and g(a/ax,O/Oy) ~- O, then x + /y

is a holomorphic coordinate on M, i.e., J(a/Ox) -- a/Oy. Thus the problem of finding

a complex structure on M compatible with the given almost complex str'acture Y is

equivalent to finding real local coordinates (x,y) such that

g(a/ax ,a /ax) - g(a/ay,a/ay) and g(a/ax ,a /ay) -- O, i.e., so called "isothermal coordi-

nates". This problem can always be solved, for any C ® Riemannian metric g (partial

differential equation methods). In our setting, thi~ solvability corresponds to the fact

that an almost complex structure on a manifold M with dimRM = 2 is always integr-

able.

There are other conditions that are equivalent to integrability of an almost complex

strttctttre. One of the more useful of these is as follows:

Define a tensor field N by

N ( X , Y ) = 2{[JXjY] - [X,Y] - J[X,JY] - J[JX,YI},

X, Y real vector fields. A calculation shows that N is function-linear in X and Y

and hence that N is a (pointwise) tensor. If there is a complex J-holomorphic coordi-

nate system defined in a neighborhood of p ~ M (i.e., a coordinate system

.q + V-Z--il Yl . . . . . x . + V - - - i y . ~ , l (a/axi ) -_ a/ay.i), then

N:,=O

because by inspection N has 0 components in the given coordinates. Thus the J of a

complex structure has N ~ 0.

287

Another condition that we could impose on J has to do with "vector fields of type

(1,0) or type (0,1)". We define a C-vector field. (i.e., a vector field X with

Xp ~ Mp ,~ C for each p ~ M) to be of type

(1,o) if i x = V-- x

(0,1) if sx=-V-Z lx.

The condition for the integrabiliry of J becomes that type (1,0) vector fields are a Lie

subalgebra, i.e. [Z,W] is type (1,0)if z and W are of type (1,0). Since

[Z,W] = [Z,W], this condition is equivalent to type (0,1) vector fields being a Lie

subalgebra. It can be shown that either of these (trivially) equivalent conditions, as well

as N ---- 0, is equivalent to integrability in our previous sense and hence equivalent to

the existence of an associated complex structure.

References

The journal literature of complex differential geometry is far too large to admit a

quick summary or even a brief representative sampling. The following books will enable

the reader to continue further with the topics introduced here, and will provide an intro-

duction to the remainder of the literature.

Items [4] and [6] cover in detail the basic results about holomorphic functions of

several complex variables, and a great deal beyond the basic results.

Chapter IX of item [5] is a treatment of basic complex differential geometry much

along the lines of the discussion here; an appendix to [5] contains the proof of the

integribility theorem for complex structures in the real analytic case; [5] also has a con-

siderable bibliography up to its publication date. Item [2] gives a treatment of complex

manifolds with a somewhat different emphasis involving a more cohomological viewpoint

including characteristic classes, etc.

Item [9], though not specifically directed toward complex differential geometry, will

288

provide some indications of recent developments in the field.

Item [I] provides a dctailcd trcatmcnt of the most important rcccnt dcvclopmcnt in

the field, S. T. Yau's proof of the Calabi conjecture.

The algebraic geometric viewpoint on complex manifolds is discussed extensively in

[3]; this includes the proof of the fundamental Kodaira theorems mentioned here in §11.

The Kodaira theorems are also treated in [7] and [8].

1. B. Bergery, J. P. Bourgnignon, E. Calabi, S. T. Yau, et. al., Premiere Classe de

Chern et Courbure de Ricci: Preuve de la Conjecture de Calabi. Seminaire Palaiseau

1978. Societe Mathematique de France, Asterisque 58. Paris. 1978.

2. S. S. Chern, Complex Manifolds without Potential Theory, second edition, Springer-

Vcrlag, New York, 1979.

3. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley, New York,

1978.

4. L. HSrmander, An Introduction to Complex Analysis in Several Variables, second edi-

tion, North Holland-American Elsevier, New York, 1973.

5. S. Kobayashi and K. NomiTu, Foundations of Differential Geometry, John Wiley,

New York, 1969.

6. S. Krantz, Function Theory of Several Complex Variables, John Wiley, New York,

1982.

7. J. Morrow and K. Kodalra, Complex Manifolds, Holt, Rinehart, Winston. New

York, 1971.

8. R. Wells, Differential Analysis on Complex Manifolds, second edition, Springer-

Verlag. New York, 1980.

9. S.T. Yau, Seminar on Differential Geometry, Princeton University Press, Annals of

Mathematics Studies, No. 102, Princeton. 1982.