of M is the bundle

TM — C

“

M

with ¬ber (T M — C)p = Tp M — C at p ∈ M . If

Tp M is a 2n-dimensional vector space over R , then

Tp M — C is a 2n-dimensional vector space over C .

We may extend J linearly to T M — C:

J(v — c) = Jv — c , v ∈ TM , c∈C.

Since J 2 = ’Id, on the complex vector space (T M — C)p , the linear map Jp has

eigenvalues ±i. Let

= {v ∈ T M — C | Jv = +iv} = (+i)-eigenspace of J

T1,0

= {v — 1 ’ Jv — i | v ∈ T M }

= (J-)holomorphic tangent vectors ;

= {v ∈ T M — C | Jv = ’iv} = (’i)-eigenspace of J

T0,1

= {v — 1 + Jv — i | v ∈ T M }

= (J-)anti-holomorphic tangent vectors .

Since

π1,0 : T M — C ’’ T1,0

v ’’ 1 (v — 1 ’ Jv — i)

2

is a (real) bundle isomorphism such that π1,0 —¦ J = iπ1,0 , and

π0,1 : T M — C ’’ T0,1

v ’’ 1 (v — 1 + Jv — i)

2

is also a (real) bundle isomorphism such that π0,1 —¦ J = ’iπ0,1 , we conclude

that we have isomorphisms of complex vector bundles

(T M, J) T1,0 T0,1 ,

where T0,1 denotes the complex conjugate bundle of T0,1 . Extending π1,0 and

π0,1 to projections of T M — C, we obtain an isomorphism

(π1,0 , π0,1 ) : T M — C ’’ T1,0 • T0,1 .

78

14.2 Forms of Type ( , m) 79

Similarly, the complexi¬ed cotangent bundle splits as

(π 1,0 , π 0,1 ) : T — M — C ’’ T 1,0 • T 0,1

where

T 1,0 = (T1,0 )— = {· ∈ T — — C | ·(Jω) = i·(ω) , ∀ω ∈ T M — C}

= {ξ — 1 ’ (ξ —¦ J) — i | ξ ∈ T — M }

= complex-linear cotangent vectors ,

T 0,1 = (T0,1 )— = {· ∈ T — — C | ·(Jω) = ’i·(ω) , ∀ω ∈ T M — C}

= {ξ — 1 + (ξ —¦ J) — i | ξ ∈ T — M }

= complex-antilinear cotangent vectors ,

and π 1,0 , π 0,1 are the two natural projections

π 1,0 : T — M — C ’’ T 1,0

· ’’ · 1,0 := 1 (· ’ i· —¦ J) ;

2

π 0,1 : T — M — C ’’ T 0,1

1

· ’’ · 0,1 := 2 (· + i· —¦ J) .

Forms of Type ( , m)

14.2

For an almost complex manifold (M, J), let

„¦k (M ; C) := sections of Λk (T — M — C)

= complex-valued k-forms on M, where

Λk (T — M — C) := Λk (T 1,0 • T 0,1 )

= • +m=k (Λ T 1,0 ) § (Λm T 0,1 )

,m (de¬nition)

Λ

,m

=• +m=k Λ .

In particular, Λ1,0 = T 1,0 and Λ0,1 = T 0,1 .

De¬nition 14.1 The di¬erential forms of type ( , m) on (M, J) are the

sections of Λ ,m :

„¦ ,m := sections of Λ ,m .

Then

„¦k (M ; C) = • ,m

+m=k „¦ .

Let π ,m : Λk (T — M — C) ’ Λ ,m be the projection map, where + m = k. The

usual exterior derivative d composed with these projections induces di¬erential

¯

operators ‚ and ‚ on forms of type ( , m):

+1,m +1,m

,m

—¦d :„¦ (M ) ’’ „¦

‚ := π (M )

¯ ,m+1 ,m ,m+1

—¦d :„¦ (M ) ’’ „¦

‚ := π (M ) .

80 14 DOLBEAULT THEORY

,m

(M ), with k = + m, then dβ ∈ „¦k+1 (M ; C):

If β ∈ „¦

¯

π r,s dβ = π k+1,0 dβ + · · · + ‚β + ‚β + · · · + π 0,k+1 dβ .

dβ =

r+s=k+1

14.3 J-Holomorphic Functions

Let f : M ’ C be a smooth complex-valued function on M . The exterior

derivative d extends linearly to C-valued functions as df = d(Ref ) + i d(Imf ).

De¬nition 14.2 A function f is (J-)holomorphic at x ∈ M if dfp is com-

plex linear, i.e., dfp —¦ J = i dfp . A function f is (J-)holomorphic if it is

holomorphic at all p ∈ M .

Exercise. Show that

1,0 0,1

dfp —¦ J = i dfp ⇐’ dfp ∈ Tp ⇐’ πp dfp = 0 .

™¦

De¬nition 14.3 A function f is (J-)anti-holomorphic at p ∈ M if dfp is

complex antilinear, i.e., dfp —¦ J = ’i dfp .

Exercise.

0,1 1,0

dfp —¦ J = ’i dfp ⇐’ dfp ∈ Tp ⇐’ πp dfp = 0

¯ 0,1 ¯

1,0

⇐’ d fp ∈ Tp ⇐’ πp dfp = 0

¯

⇐’ f is holomorphic at p ∈ M .

™¦

¯

De¬nition 14.4 On functions, d = ‚ + ‚, where

¯

‚ := π 1,0 —¦ d ‚ := π 0,1 —¦ d .

and

Then

¯

⇐’

f is holomorphic ‚f = 0 ,

⇐’

f is anti-holomorphic ‚f = 0 .

“ What about higher di¬erential forms?

81

14.4 Dolbeault Cohomology

14.4 Dolbeault Cohomology

¯

Suppose that d = ‚ + ‚, i.e.,

¯ ,m

∀β ∈ „¦

dβ = ‚β + ‚β , .

+1,m ,m+1

∈„¦ ∈„¦

,m

Then, for any form β ∈ „¦ ,

¯ ¯ ¯

0 = d2 β = ‚ 2 β + ‚ ‚β + ‚‚β + ‚ 2 β ,

+2,m +1,m+1 ,m+2

∈„¦ ∈„¦ ∈„¦

which implies ±2

¯

‚ =0

¯¯

‚ ‚ + ‚‚ = 0

2

‚ =0

¯

Since ‚ 2 = 0, the chain

¯ ¯ ¯

‚ ‚ ‚

,0 ,1 ,2

0 ’’ „¦ ’’ „¦ ’’ „¦ ’’ · · ·

is a di¬erential complex; its cohomology groups

¯

ker ‚ : „¦ ,m ’’ „¦ ,m+1

,m

HDolbeault (M ) := ¯

im ‚ : „¦ ,m’1 ’’ „¦ ,m

are called the Dolbeault cohomology groups.

„¦0,0

d

‚ d‚¯

d

d

© ‚

d

„¦1,0 „¦0,1

d d

‚ d‚ ‚ d‚

¯ ¯

d d

d d

© ‚

d

© ‚

d

„¦2,0 „¦1,1 „¦0,2

. . .

. . .

. . .

¯

“ When is d = ‚ + ‚? See the next lecture.

Homework 10: Integrability

This set of problems is from [11, p.46-47].

1. Let (M, J) be an almost complex manifold. Its Nijenhuis tensor N is:

N (v, w) := [Jv, Jw] ’ J[v, Jw] ’ J[Jv, w] ’ [v, w] ,

where v and w are vector ¬elds on M , [·, ·] is the usual bracket

[v, w] · f := v · (w · f ) ’ w · (v · f ) , for f ∈ C ∞ (M ) ,

and v · f = df (v).

(a) Check that, if the map v ’ [v, w] is complex linear (in the sense that

it commutes with J), then N ≡ 0.

(b) Show that N is actually a tensor, that is: N (v, w) at x ∈ M depends

only on the values vx , wx ∈ Tx M and not really on the vector ¬elds

v and w.

(c) Compute N (v, Jv). Deduce that, if M is a surface, then N ≡ 0.

A theorem of Newlander and Nirenberg [88] states that an almost complex

manifold (M, J) is a complex (analytic) manifold if and only if N ≡ 0.

Combining (c) with the fact that any orientable surface is symplectic,

we conclude that any orientable surface is a complex manifold, a result

already known to Gauss.

2. Let N be as above. For any map f : R2n ’ C and any vector ¬eld v on

R2n , we have v · f = v · (f1 + if2 ) = v · f1 + i v · f2 , so that f ’ v · f is a

complex linear map.

(a) Let R2n be endowed with an almost complex structure J, and suppose

that f is a J-holomorphic function, that is,

df —¦ J = i df .

Show that df (N (v, w)) = 0 for all vector ¬elds v, w.

(b) Suppose that there exist n J-holomorphic functions, f1 , . . . , fn , on

R2n , which are independent at some point x, i.e., the real and imagi-

nary parts of (df1 )x , . . . , (dfn )x form a basis of Tx R2n . Show that N

—

vanishes identically at x.

(c) Assume that M is a complex manifold and J is its complex structure.

Show that N vanishes identically everywhere on M .

In general, an almost complex manifold has no J-holomorphic functions

at all. On the other hand, it has plenty of J-holomorphic curves: maps

f : C ’ M such that df —¦ i = J —¦ df . J-holomorphic curves, also known as

pseudo-holomorphic curves, provide a main tool in symplectic topol-

ogy, as ¬rst realized by Gromov [49].

82

Part VI

K¨hler Manifolds

a

K¨hler geometry lies at the intersection of complex, riemannian and symplec-

a

tic geometries, and plays a central role in all of these ¬elds. We will start by

reviewing complex manifolds. After describing the local normal form for K¨h- a

ler manifolds (Lecture 16), we conclude with a summary of Hodge theory for

compact K¨hler manifolds (Lecture 17).

a

15 Complex Manifolds

15.1 Complex Charts

De¬nition 15.1 A complex manifold of (complex) dimension n is a set M

with a complete complex atlas

A = {(U± , V± , •± ) , ± ∈ index set I}

where M = ∪± U± , the V± ™s are open subsets of Cn , and the maps •± : U± ’

V± are such that the transition maps ψ±β are biholomorphic as maps on open

subsets of Cn :

U± © U β

d

•± d •β

d

d

© ‚

d

’1

ψ±β = •β —¦ •± E

V±β Vβ±

where V±β = •± (U± © Uβ ) ⊆ Cn and Vβ± = •β (U± © Uβ ) ⊆ Cn . ψ±β being

’1

biholomorphic means that ψ±β is a bijection and that ψ±β and ψ±β are both

holomorphic.

Proposition 15.2 Any complex manifold has a canonical almost complex struc-

ture.

Proof.

1) Local de¬nition of J:

Let (U, V, • : U ’ V) be a complex chart for a complex manifold M with

• = (z1 , . . . , zn ) written in components relative to complex coordinates

zj = xj + iyj . At p ∈ U

‚ ‚

Tp M = R-span of , : j = 1, . . . , n .

‚xj ‚yj

p p

83

84 15 COMPLEX MANIFOLDS

De¬ne J over U by

‚ ‚

Jp =

‚xj ‚yj

p p

j = 1, . . . , n .

‚ ‚

=’

Jp

‚yj ‚xj

p p

2) This J is well-de¬ned globally:

If (U, V, •) and (U , V , • ) are two charts, we need to show that J = J

on their overlap.

On U © U , ψ —¦ • = • . If zj = xj + iyj and wj = uj + ivj are coordinates

on U and U , respectively, so that • and • can be written in components

• = (z1 , . . . , zn ), • = (w1 , . . . , wn ), then ψ(z1 , . . . , zn ) = (w1 , . . . , wn ).

Taking the derivative of a composition

±

‚ ‚uj ‚ ‚vj ‚

= +

‚x

‚xk ‚uj ‚xk ‚vj

k j

‚ ‚uj ‚ ‚vj ‚

= +

‚yk ‚yk ‚uj ‚yk ‚vj

j

Since ψ is biholomorphic, each component of ψ satis¬es the Cauchy-

Riemann equations:

±

‚uj = ‚vj

‚x

‚yk

k

j, k = 1, . . . , n .

‚u

‚vj

j

=’

‚yk ‚xk

These equations imply

‚uj ‚ ‚vj ‚ ‚uj ‚ ‚vj ‚

J + = +

‚xk ‚uj ‚xk ‚vj ‚yk ‚uj ‚yk ‚vj

j j

«

¬ ·

¬ ‚u ‚ ‚vj ‚ ·

¬j ·

’

¬ ·

¬ ‚xk ‚vj ‚xj ‚uj ·

j

‚vj ‚u

’ ‚y j

‚yk

k

which matches the equation

‚ ‚

J = .

‚xk ‚yk

85

15.2 Forms on Complex Manifolds

15.2 Forms on Complex Manifolds

Suppose that M is a complex manifold and J is its canonical almost complex

structure. What does the splitting „¦k (M ; C) = • +m=k „¦ ,m look like? ([22,

48, 65, 107] are good references for this material.)

Let U ⊆ M be a coordinate neighborhood with complex coordinates z1 , . . . , zn ,

zj = xj + iyj , and real coordinates x1 , y1 , . . . , xn , yn . At p ∈ U,

‚ ‚

Tp M = R-span ,

‚xj ‚yj

p p

‚ ‚

Tp M — C = C-span ,

‚xj ‚yj

p p

1 ‚ ‚ 1 ‚ ‚

’i •

= C-span +i

C-span

2 ‚xj ‚yj 2 ‚xj ‚yj

p p p p

T1,0 = (+i)-eigenspace of J T0,1 = (’i)-eigenspace of J

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

’ i ‚yj ’ i ‚yj = ’i

J =i J + i ‚yj + i ‚yj

‚xj ‚xj ‚xj ‚xj

This can be written more concisely using:

De¬nition 15.3

‚ 1 ‚ ‚ ‚ 1 ‚ ‚

’i

:= and := +i .

‚zj 2 ‚xj ‚yj ‚ zj

¯ 2 ‚xj ‚yj

Hence,

‚ ‚

(T1,0 )p = C-span : j = 1, . . . , n , (T0,1 )p = C-span : j = 1, . . . , n .

‚zj ‚ zj

¯

p p

Similarly,

T — M — C = C-span{dxj , dyj : j = 1, . . . , n}

= C-span{dxj + idyj : j = 1, . . . , n} • C-span{dxj ’ idyj : j = 1, . . . , n}

T 1,0 T 0,1

(dxj + idyj ) —¦ J = i(dxj + idyj ) (dxj ’ idyj ) —¦ J = ’i(dxj ’ idyj )

Putting

d¯j = dxj ’ idyj ,

dzj = dxj + idyj and z

we obtain

T 1,0 = C-span{dzj : j = 1, . . . , n} , T 0,1 = C-span{d¯j : j = 1, . . . , n} .

z

86 15 COMPLEX MANIFOLDS

On the coordinate neighborhood U,

| bj ∈ C ∞ (U; C)

(1, 0)-forms = j bj dzj

| bj ∈ C ∞ (U; C)

(0, 1)-forms = j bj d¯j

z

bj1 ,j2 dzj1 § dzj2 | bj1 ,j2 ∈ C ∞ (U; C)

(2, 0)-forms = j1 <j2

bj1 ,j2 dzj1 § d¯j2 | bj1 ,j2 ∈ C ∞ (U; C)

(1, 1)-forms = z

j1 ,j2

bj1 ,j2 d¯j1 § d¯j2 | bj1 ,j2 ∈ C ∞ (U; C)

(0, 2)-forms = z z

j1 <j2

If we use multi-index notation:

1 ¤ j1 < . . . < jm ¤ n

J = (j1 , . . . , jm )

|J| = m

dzL = dzj1 § dzj2 § . . . § dzjm

then

±

bJ,K dzJ § d¯K | bJ,K ∈ C ∞ (U; C)

,m

„¦ = ( , m)-forms = z .

|J|= ,|K|=m

15.3 Di¬erentials

On a coordinate neighborhood U, a form β ∈ „¦k (M ; C) may be written as

with aJ,K ∈ C ∞ (U; C) .

aJ,K dxJ § dyK ,

β=

|J|+|K|=k

We would like to know whether the following equality holds:

?

¯ ¯

(‚aJ,K + ‚aJ,K )dxJ § dyK = (‚ + ‚) aJ,K dxJ § dyK .

dβ =

If we use the identities

1

dxj + idyj = dzj dxj = 2 (dzj + d¯j )

z

⇐’ 1

dxj ’ idyj = d¯j dyj = 2i (dzj ’ d¯j )

z z

87

15.3 Di¬erentials

after substituting and reshu¬„ing, we obtain

bJ,K dzJ § d¯K

β = z

|J|+|K|=k

«

bJ,K dzJ § d¯K ,

= z

+m=k |J|= ,|K|=m

,m

∈„¦

«

dbJ,K § dzJ § d¯K

dβ = z

+m=k |J|= ,|K|=m

¯

‚bJ,K + ‚bJ,K § dzJ § d¯K

= z

+m=k |J|= ,|K|=m

¯

(because d = ‚ + ‚ on functions)

«

¬ ·

¬ ·

¬ ·

¯

‚bJ,K § dzJ § d¯K + § dzJ § d¯K ·

= z ‚bJ,K z

¬

¬ ·

+m=k |J|=

|J|= ,|K|=m

,|K|=m

+1,m ,m+1

∈„¦ ∈„¦

¯

= ‚β + ‚β .

¯

Therefore, d = ‚ + ‚ on forms of any degree for a complex manifold.

¯

Conclusion. If M is a complex manifold, then d = ‚ + ‚. (For an almost

complex manifold this fails because there are no coordinate functions zj to give

a suitable basis of 1-forms.)

Remark. If b ∈ C ∞ (U; C), in terms of z and z , we obtain the following formu-

¯

las:

‚b ‚b

db = dxj + dyj

‚xj ‚yj

j

1 ‚b ‚b 1 ‚b ‚b

’i (dxj ’ idyj )

= (dxj + idyj ) + +i

2 ‚xj ‚yj 2 ‚xj ‚yj

j

‚b ‚b

= dzj + d¯j

z .

‚zj ‚ zj

¯

j

Hence:

‚b = π 1,0 db = ‚b

j ‚zj dzj

¯

‚b = π 0,1 db = ‚b

j ‚ zj d¯j

¯z

88 15 COMPLEX MANIFOLDS

™¦

,m

In the case where β ∈ „¦ , we have

¯

dβ = ‚β + ‚β = ( + 1, m)-form + ( , m + 1)-form

0 = d2 β = ( + 2, m)-form + ( + 1, m + 1)-form + ( , m + 2)-form

¯¯ ¯

‚ 2 β + (‚ ‚ + ‚‚)β + ‚ 2 β .

=

0 0

0

¯

Hence, ‚ 2 = 0.

The Dolbeault theorem states that for complex manifolds

,m

HDolbeault(M ) = H m (M ; O(„¦( ,0)

)) ,

where O(„¦( ,0)

) is the sheaf of forms of type ( , 0) over M .

Theorem 15.4 (Newlander-Nirenberg, 1957 [88])

Let (M, J) be an almost complex manifold. Let N be the Nijenhuis tensor

(de¬ned in Homework 10). Then:

⇐’

M is a complex manifold J is integrable

⇐’ N ≡0

¯

⇐’ d=‚+‚

¯

‚2 = 0

⇐’

π 2,0 d|„¦0,1 = 0 .

⇐’

For the proof of this theorem, besides the original reference, see also [22, 30,

¯

48, 65, 107]. Naturally most almost complex manifolds have d = ‚ + ‚.

Homework 11: Complex Projective Space

The complex projective space CPn is the space of complex lines in Cn+1 :

CPn is obtained from Cn+1 \{0} by making the identi¬cations (z0 , . . . , zn ) ∼

(»z0 , . . . , »zn ) for all » ∈ C \ {0}. One denotes by [z0 , . . . , zn ] the equivalence

class of (z0 , . . . , zn ), and calls z0 , . . . , zn the homogeneous coordinates of the

point p = [z0 , . . . , zn ]. (The homogeneous coordinates are, of course, only de-

termined up to multiplication by a non-zero complex number ».)

Let Ui be the subset of CPn consisting of all points p = [z0 , . . . , zn ] for which

zi = 0. Let •i : Ui ’ Cn be the map

zi’1 zi+1

z0 zn

•i ([z0 , . . . , zn ]) = zi , . . . , zi , zi , . . . , zi .

1. Show that the collection

{(Ui , Cn , •i ), i = 0, . . . , n}

is an atlas in the complex sense, i.e., the transition maps are biholomorphic.

Conclude that CPn is a complex manifold.

Work out the transition maps associated with (U0 , Cn , •0 ) and

Hint:

n , • ). Show that the transition diagram has the form

(U1 , C 1

U0 © U 1

d•

•0 d1

d

© d

‚

E V1,0

•0,1

V0,1

where V0,1 = V1,0 = {(z1 , . . . , zn ) ∈ Cn | z1 = 0} and

z2

, . . . , zn )

1

•0,1 (z1 , . . . , zn ) = ( z , .

z1 z1

1

2. Show that the 1-dimensional complex manifold CP1 is di¬eomorphic, as a

real 2-dimensional manifold, to S 2 .

Stereographic projection.

Hint:

89

16 K¨hler Forms

a

16.1 K¨hler Forms

a

De¬nition 16.1 A K¨hler manifold is a symplectic manifold (M, ω) equipped

a

with an integrable compatible almost complex structure. The symplectic form ω

is then called a K¨hler form.

a

It follows immediately from the previous de¬nition that

(M, ω) is K¨hler =’ M is a complex manifold

a

„¦k (M ; C) = • ,m

+m=k „¦

=’ ¯

d=‚+‚

where

+1,m +1,m

,m

—¦d:„¦ ’„¦

‚=π

¯ ,m+1 ,m ,m+1

—¦d:„¦ ’„¦

‚=π .

On a complex chart (U, z1 , . . . , zn ), n = dimC M ,

±

∞

,m

bJK dzJ § d¯K | bJK ∈ C (U; C) ,

„¦ = z

|J|= ,|K|=m

where

= dzj1 § . . . § dzj ,

J = (j1 , . . . , j ) , j1 < . . . < j , dzJ

= d¯k1 § . . . § d¯km .

K = (k1 , . . . , km ) , k1 < . . . < k m , d¯K

z z z

On the other hand,

(M, ω) is K¨hler =’ ω is a symplectic form .

a

“ Where does ω ¬t with respect to the above decomposition?

A K¨hler form ω is

a

1. a 2-form,

2. compatible with the complex structure,

3. closed,

4. real-valued, and

5. nondegenerate.

These properties translate into:

90

91

16.1 K¨hler Forms

a

1. „¦2 (M ; C) = „¦2,0 • „¦1,1 • „¦0,2 .

On a local complex chart (U, z1 , . . . , zn ),

ajk dzj § dzk + bjk dzj § d¯k + cjk d¯j § d¯k

ω= z z z

for some ajk , bjk , cjk ∈ C ∞ (U; C).

2. J is a symplectomorphism, that is, J — ω = ω where (J — ω)(u, v) := ω(Ju, Jv).

J — dzj = dzj —¦ J = idzj

J — d¯j = d¯j —¦ J = ’id¯j

z z z

’1

1

’1

J —ω bjk dzj § d¯k + (’i)2

(i · i) ajk dzj § dzk + i(’i) cjk d zj § d¯k

= z ¯ z

J —ω ω ∈ „¦1,1 .

⇐’ ⇐’

=ω ajk = 0 = cjk , all j, k

‚ω = 0 ω is ‚-closed

¯

3. 0 = dω = ‚ω + ‚ω =’ ¯ ¯

‚ω = 0 ω is ‚-closed

(2,1)’form (1,2)’form

Hence, ω de¬nes a Dolbeault (1, 1) cohomology class,

1,1

[ω] ∈ HDolbeault (M ) .

i

Putting bjk = 2 hjk ,

n

i

hjk ∈ C ∞ (U; C).

hjk dzj § d¯k ,

ω= z

2

j,k=1

4. ω real-valued ⇐’ ω = ω.

i i i

ω=’ hjk d¯j § dzk = hjk dzk § d¯j = hkj dzj § d¯k

z z z

2 2 2

⇐’ hjk = hkj ,

ω real

i.e., at every point p ∈ U, the n — n matrix (hjk (p)) is hermitian.

5. nondegeneracy: ω n = ω § . . . § ω = 0.

n

Exercise. Check that

n

i

n

det(hjk ) dz1 § d¯1 § . . . § dzn § d¯n .

ω = n! z z

2

™¦

Now

⇐’ detC (hjk ) = 0 ,

ω nondegenerate

i.e., at every p ∈ M , (hjk (p)) is a nonsingular matrix.

¨

92 16 KAHLER FORMS

2. Again the positivity condition: ω(v, Jv) > 0, ∀v = 0.

™¦

Exercise. Show that (hjk (p)) is positive-de¬nite.

ω positive ⇐’ (hjk ) 0,

i.e., at each p ∈ U, (hjk (p)) is positive-de¬nite.

¯

Conclusion. K¨hler forms are ‚- and ‚-closed (1, 1)-forms, which are given on

a

a local chart (U, z1 , . . . , zn ) by

n

i

hjk dzj § d¯k

ω= z

2

j,k=1

where, at every point p ∈ U, (hjk (p)) is a positive-de¬nite hermitian matrix.

16.2 An Application

Theorem 16.2 (Banyaga) Let M be a compact complex manifold. Let ω0

2

and ω1 be K¨hler forms on M . If [ω0 ] = [ω1 ] ∈ HdeRham (M ), then (M, ω0 ) and

a

(M, ω1 ) are symplectomorphic.

Proof. Any combination ωt = (1 ’ t)ω0 + tω1 is symplectic for 0 ¤ t ¤ 1,

because, on a complex chart (U, z1 , . . . , zn ), where n = dimC M , we have

i

h0 dzj § d¯k

ω0 = z

jk

2

i

h1 dzj § d¯k

ω1 = z

jk

2

i

where ht = (1 ’ t)h0 + th1 .

ht dzj § d¯k ,

ωt = z

jk jk jk jk

2

(h0 ) 0 , (h1 ) 0 =’ (ht ) 0.

jk jk jk

Apply the Moser theorem (Theorem 7.2).

16.3 Recipe to Obtain K¨hler Forms

a

De¬nition 16.3 Let M be a complex manifold. A function ρ ∈ C ∞ (M ; R) is

strictly plurisubharmonic (s.p.s.h.) if, on each local complex chart (U, z 1 , . . . , zn ),

‚2ρ

is positive-de¬nite at all p ∈ U.

where n = dimC M , the matrix ‚zj ‚ zk (p)

¯

Proposition 16.4 Let M be a complex manifold and let ρ ∈ C ∞ (M ; R) be

s.p.s.h.. Then

i¯

ω = ‚ ‚ρ is K¨hler .

a

2

93

16.3 Recipe to Obtain K¨hler Forms

a

A function ρ as in the previous proposition is called a (global) K¨hler

a

potential.

Proof. Simply observe that:

± 0

‚ω ¯

‚ 2 ‚ρ = 0

i

= 2

¯

‚ω ¯¯ ¯

‚‚ ‚ρ = ’ 2 ‚ ‚ 2 ρ = 0

i i

=

2

¯ 0

’‚ ‚

¯

dω = ‚ω + ‚ω = 0 =’ ω is closed .

i¯ i¯

ω = ’ 2 ‚‚ρ = 2 ‚ ‚ρ = ω =’ ω is real .

ω ∈ „¦1,1 =’ J — ω = ω =’ ω(·, J·) is symmetric .

Exercise. Show that, for f ∈ C ∞ (U; C),

‚f ‚f

¯

‚f = dzj and ‚f = d¯j .

z

‚zj ‚ zj

¯

Since the right-hand sides are in „¦1,0 and „¦0,1 , respectively, it su¬ces to show

™¦

that the sum of the two expressions is df .

‚2ρ

i¯ i ‚ ‚ρ i

dzj § d¯k = dzj § d¯k .

ω = ‚ ‚ρ = z z

2 2 ‚zj ‚ zk

¯ 2 ‚zj ‚ zk

¯

hjk

ρ is s.p.s.h =’ (hjk ) 0 =’ ω(·, J·) is positive .

In particular, ω is nondegenerate.

R2n , with complex coordinates (z1 , . . . , zn ) and

Example. Let M = Cn

corresponding real coordinates (x1 , y1 , . . . , xn , yn ) via zj = xj + iyj . Let

n

(x2 + yj ) =

2

|zj |2 =

ρ(x1 , y1 , . . . , xn , yn ) = zj z j .

¯

j

j=1

Then

‚ ‚ρ ‚

= zk = δjk ,

‚zj ‚ zk

¯ ‚zj

so

‚2ρ

(hjk ) = = (δjk ) = Id 0 =’ ρ is s.p.s.h. .

‚zj ‚ zk

¯

¨

94 16 KAHLER FORMS

The corresponding K¨hler form

a

i¯ i

δjk dzj § d¯k

ω = 2 ‚ ‚ρ = z

2

j,k

i

dzj § d¯j = dxj § dyj

= z is the standard form .

2

j j

™¦

16.4 Local Canonical Form for K¨hler Forms

a

There is a local converse to the previous construction of K¨hler forms.

a

Theorem 16.5 Let ω be a closed real-valued (1, 1)-form on a complex manifold

M and let p ∈ M . Then there exist a neighborhood U of p and ρ ∈ C ∞ (U; R)

such that, on U,

i¯

ω = ‚ ‚ρ .

2

The function ρ is then called a (local) K¨hler potential.

a

The proof requires holomorphic versions of Poincar´™s lemma, namely, the

e

local triviality of Dolbeault groups:

,m

∀p ∈ M ∃ neighborhood U of p such that HDolbeault (U) = 0 , m > 0 ,

and the local triviality of the holomorphic de Rham groups; see [48].

Theorem 16.6 Let M be a complex manifold, ρ ∈ C ∞ (M ; R) s.p.s.h., X a

complex submanifold, and i : X ’ M the inclusion map. Then i— ρ is s.p.s.h..

Proof. Let dimC M = n and dimC X = n ’ m. For p ∈ X, choose a chart

(U, z1 , . . . , zn ) for M centered at p and adapted to X, i.e., X © U is given by

z1 = . . . = zm = 0. In this chart, i— ρ = ρ(0, 0, . . . , 0, zm+1 , . . . , zn ).

‚2ρ

—

i p is s.p.s.h. ⇐’ (0, . . . , 0, zm+1 , . . . , zn ) is positive-de¬nite ,

‚zm+j ‚ zm+k

¯

‚2

which holds since this is a minor of ‚zj ‚ zk (0, . . . , 0, zm+1 , . . . , zn ) .

¯

Corollary 16.7 Any complex submanifold of a K¨hler manifold is also K¨hler.

a a

De¬nition 16.8 Let (M, ω) be a K¨hler manifold, X a complex submanifold,

a

and i : X ’ M the inclusion. Then (X, i— ω) is called a K¨hler submanifold.

a

95

16.4 Local Canonical Form for K¨hler Forms

a

i

Example. Complex vector space (Cn , ω) where ω = dzj § d¯j is K¨hler.

z a

2

Every complex submanifold of Cn is K¨hler. ™¦

a

Example. The complex projective space is

CPn = Cn+1 \{0}/ ∼

where

(z0 , . . . , zn ) ∼ (»z0 , . . . , »zn ) , » ∈ C\{0} .

The Fubini-Study form (see Homework 12) is K¨hler. Therefore, every non-

a

singular projective variety is a K¨hler submanifold. Here we mean

a

non-singular = smooth

projective variety = zero locus of a collection

of homogeneous polynomials .

™¦

Homework 12: The Fubini-Study Structure

The purpose of the following exercises is to describe the natural K¨hler structure

a

n

on complex projective space, CP .

1. Show that the function on Cn

z ’’ log(|z|2 + 1)

is strictly plurisubharmonic. Conclude that the 2-form

i¯

ωFS = 2 ‚ ‚ log(|z|2 + 1)

is a K¨hler form. (It is usually called the Fubini-Study form on Cn .)

a

Hint: A hermitian n — n matrix H is positive de¬nite if and only if v — Hv > 0

for any v ∈ Cn \ {0}, where v — is the transpose of the vector v . To prove

¯

positive-de¬niteness, either apply the Cauchy-Schwarz inequality, or use the

following symmetry observation: U(n) acts transitively on S 2n’1 and ωFS is

U(n)-invariant, thus it su¬ces to show positive-de¬niteness along one direction.

2. Let U be the open subset of Cn de¬ned by the inequality z1 = 0, and let

• : U ’ U be the map

1

•(z1 , . . . , zn ) = z1 (1, z2 , . . . , zn ) .

Show that • maps U biholomorphically onto U and that

•— log(|z|2 + 1) = log(|z|2 + 1) + log |z1|2 . ()

1

3. Notice that, for every point p ∈ U, we can write the second term in ( ) as

the sum of a holomorphic and an anti-holomorphic function:

’ log z1 ’ log z1

on a neighborhood of p. Conclude that

¯ ¯

‚ ‚•— log(|z|2 + 1) = ‚ ‚ log(|z|2 + 1)

and hence that •— ωFS = ωFS .

Hint: You need to use the fact that the pullback by a holomorphic map •—

¯

commutes with the ‚ and ‚ operators. This is a consequence of •— preserving

— („¦p,q ) ⊆ „¦p,q , which in turn is implied by •— dz = ‚• ⊆ „¦1,0

form type, • j j

¯

and •— dzj = ‚•j ⊆ „¦0,1 , where •j is the jth component of • with respect to

local complex coordinates (z1 , . . . , zn ).

4. Recall that CPn is obtained from Cn+1 \ {0} by making the identi¬ca-

tions (z0 , . . . , zn ) ∼ (»z0 , . . . , »zn ) for all » ∈ C \ {0}; [z0 , . . . , zn ] is the

equivalence class of (z0 , . . . , zn ).

96

97

HOMEWORK 12

For i = 0, 1, . . . , n, let

Ui = {[z0 , . . . , zn ] ∈ CPn |zi = 0}

zi’1 zi+1

z0 zn

•i : Ui ’ C n •i ([z0 , . . . , zn ]) = zi , . . . , zi , zi , . . . , zi .

Homework 11 showed that the collection {(Ui , Cn , •i ), i = 0, . . . , n} is a

complex atlas (i.e., the transition maps are biholomorphic). In particular,

it was shown that the transition diagram associated with (U0 , Cn , •0 ) and

(U1 , Cn , •1 ) has the form

U0 © U 1

d

d •1

•0

d

d

© ‚

d

•0,1 E V1,0

V0,1

where V0,1 = V1,0 = {(z1 , . . . , zn ) ∈ Cn | z1 = 0} and •0,1 (z1 , . . . , zn ) =

( z1 , z1 , . . . , zn ). Now the set U in exercise 2 is equal to the sets V0,1 and

z2

z1

1

V1,0 , and the map • coincides with •0,1 .

Show that •— ωFS and •— ωFS are identical on the overlap U0 © U1 .

0 1

More generally, show that the K¨hler forms •— ωFS “glue together” to

a i

n

de¬ne a K¨hler structure on CP . This is called the Fubini-Study form

a

on complex projective space.

5. Prove that for CP1 the Fubini-Study form on the chart U0 = {[z0 , z1 ] ∈

CP1 |z0 = 0} is given by the formula

dx § dy

ωFS =

(x2 + y 2 + 1)2

z1

where = z = x + iy is the usual coordinate on C.

z0

6. Compute the total area of CP1 = C ∪ {∞} with respect to ωFS :

dx § dy

ωFS = .

(x2 + y 2 + 1)2

CP1 R2

7. Recall that CP1 S 2 as real 2-dimensional manifolds (Homework 11).

On S 2 there is the standard area form ωstd induced by regarding it as the

unit sphere in R3 (Homework 6): in cylindrical polar coordinates (θ, h) on

S 2 away from its poles (0 ¤ θ < 2π and ’1 ¤ h ¤ 1), we have

ωstd = dθ § dh .

Using stereographic projection, show that

1

ωFS = ω.

4 std

17 Compact K¨hler Manifolds

a

17.1 Hodge Theory

Let M be a complex manifold. A K¨hler form ω on M is a symplectic form which

a

is compatible with the complex structure. Equivalently, a K¨hler form ω is a ‚-

a

¯

and ‚-closed form of type (1, 1) which, on a local chart (U, z1 , . . . , zn ) is given by

n

i

ω = 2 j,k=1 hjk dzj § d¯k , where, at each x ∈ U, (hjk (x)) is a positive-de¬nite

z

hermitian matrix. The pair (M, ω) is then called a K¨hler manifold.

a

Theorem 17.1 (Hodge) On a compact K¨hler manifold (M, ω) the Dolbeault

a

cohomology groups satisfy

,m

k

HdeRham (M ; C) HDolbeault(M ) (Hodge decomposition)

+m=k

,m

,m

H m, . In particular, the spaces HDolbeault are ¬nite-dimensional.

with H

Hodge identi¬ed the spaces of cohomology classes of forms with spaces of

actual forms, by picking the representative from each class which solves a certain

di¬erential equation, namely the harmonic representative.

(1) The Hodge —-operator.

Each tangent space V = Tx M has a positive inner product ·, · , part of

the riemannian metric in a compatible triple; we forget about the complex

and symplectic structures until part (4).

Let e1 , . . . , en be a positively oriented orthonormal basis of V .

The star operator is a linear operator — : Λ(V ) ’ Λ(V ) de¬ned by

—(1) = e1 § . . . § en

—(e1 § . . . § en ) = 1

—(e1 § . . . § ek ) = ek+1 § . . . § en .

We see that — : Λk (V ) ’ Λn’k (V ) and satis¬es —— = (’1)k(n’k) .

(2) The codi¬erential and the laplacian are the operators de¬ned by:

= (’1)n(k+1)+1 — d— : „¦k (M ) ’ „¦k’1 (M )

δ

: „¦k (M ) ’ „¦k (M ) .

∆ = dδ + δd

The operator ∆ is also called the Laplace-Beltrami operator.

‚2

n

Exercise. Check that, on „¦0 (Rn ) = C ∞ (Rn ), ∆ = ’ ™¦

i=1 ‚x2 .

i

Exercise. Check that ∆— = —∆. ™¦

Suppose that M is compact. De¬ne an inner product on forms by

·, · : „¦k — „¦k ’ R , ± § —β .

±, β =

M

98

99

17.1 Hodge Theory

Exercise. Check that this is symmetric, positive-de¬nite and satis¬es

™¦

d±, β = ±, δβ .

Therefore, δ is often denoted by d— and called the adjoint of d. (When

M is not compact, we still have a formal adjoint of d with respect to the

nondegenerate bilinear pairing ·, · : „¦k — „¦k ’ R de¬ned by a similar

c

formula, where „¦k is the space of compactly supported k-forms.) Also, ∆

c

is self-adjoint:

Exercise. Check that ∆±, β = ±, ∆β , and that ∆±, ± = |d±|2 +

|δ±|2 ≥ 0, where | · | is the norm with respect to this inner product. ™¦

(3) The harmonic k-forms are the elements of Hk := {± ∈ „¦k | ∆± = 0}.

Note that ∆± = 0 ⇐’ d± = δ± = 0. Since a harmonic form is d-closed,

it de¬nes a de Rham cohomology class.

Theorem 17.2 (Hodge) Every de Rham cohomology class on a com-

pact oriented riemannian manifold M possesses a unique harmonic repre-

sentative, i.e.,

Hk HdeRham (M ; R) .

k

In particular, the spaces Hk are ¬nite-dimensional. We also have the

following orthogonal decomposition with respect to ·, · :

„¦k Hk • ∆(„¦k (M ))

(Hodge decomposition on forms) .

Hk • d„¦k’1 • δ„¦k+1

The proof involves functional analysis, elliptic di¬erential operators, pseu-

dodi¬erential operators and Fourier analysis; see [48, 107].

So far, this was ordinary Hodge theory, considering only the metric and

not the complex structure.

(4) Complex Hodge Theory.

¯¯ ¯¯

When M is K¨hler, the laplacian satis¬es ∆ = 2(‚ ‚ — + ‚ — ‚) (see, for

a

example, [48]) and preserves the decomposition according to type, ∆ :

„¦ ,m ’ „¦ ,m . Hence, harmonic forms are also bigraded

Hk = ,m

H .

+m=k

Theorem 17.3 (Hodge) Every Dolbeault cohomology class on a com-

pact K¨hler manifold (M, ω) possesses a unique harmonic representative,

a

i.e.,

,m

H ,m HDolbeault(M )

¨

100 17 COMPACT KAHLER MANIFOLDS

,m