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Let (M, J) be an almost complex manifold. The complexi¬ed tangent bundle
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

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,

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

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 • 䄦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

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