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Let (M, J) be an almost complex manifold. The complexiп¬Ғed tangent bundle
of M is the bundle
TM вҠ— C
вҶ“
M
with п¬Ғber (T M вҠ— C)p = Tp M вҠ— C at p вҲҲ M . If

Tp M is a 2n-dimensional vector space over R , then
Tp M вҠ— C is a 2n-dimensional vector space over C .

We may extend J linearly to T M вҠ— C:

J(v вҠ— c) = Jv вҠ— c , v вҲҲ TM , cвҲҲC.

Since J 2 = вҲ’Id, on the complex vector space (T M вҠ— C)p , the linear map Jp has
eigenvalues В±i. Let

= {v вҲҲ T M вҠ— C | Jv = +iv} = (+i)-eigenspace of J
T1,0
= {v вҠ— 1 вҲ’ Jv вҠ— i | v вҲҲ T M }
= (J-)holomorphic tangent vectors ;

= {v вҲҲ T M вҠ— C | Jv = вҲ’iv} = (вҲ’i)-eigenspace of J
T0,1
= {v вҠ— 1 + Jv вҠ— i | v вҲҲ T M }
= (J-)anti-holomorphic tangent vectors .

Since
ПҖ1,0 : T M вҠ— C вҲ’вҶ’ T1,0
v вҲ’вҶ’ 1 (v вҠ— 1 вҲ’ Jv вҠ— i)
2

is a (real) bundle isomorphism such that ПҖ1,0 в—¦ J = iПҖ1,0 , and

ПҖ0,1 : T M вҠ— C вҲ’вҶ’ T0,1
v вҲ’вҶ’ 1 (v вҠ— 1 + Jv вҠ— i)
2

is also a (real) bundle isomorphism such that ПҖ0,1 в—¦ J = вҲ’iПҖ0,1 , 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 ПҖ1,0 and
ПҖ0,1 to projections of T M вҠ— C, we obtain an isomorphism

(ПҖ1,0 , ПҖ0,1 ) : T M вҠ— C вҲ’вҶ’ T1,0 вҠ• T0,1 .

78
14.2 Forms of Type ( , m) 79

Similarly, the complexiп¬Ғed cotangent bundle splits as

(ПҖ 1,0 , ПҖ 0,1 ) : T вҲ— M вҠ— C вҲ’вҶ’ T 1,0 вҠ• T 0,1

where
T 1,0 = (T1,0 )вҲ— = {О· вҲҲ T вҲ— вҠ— C | О·(JПү) = iО·(Пү) , вҲҖПү вҲҲ T M вҠ— C}
= {Оҫ вҠ— 1 вҲ’ (Оҫ в—¦ J) вҠ— i | Оҫ вҲҲ T вҲ— M }
= complex-linear cotangent vectors ,

T 0,1 = (T0,1 )вҲ— = {О· вҲҲ T вҲ— вҠ— C | О·(JПү) = вҲ’iО·(Пү) , вҲҖПү вҲҲ T M вҠ— C}
= {Оҫ вҠ— 1 + (Оҫ в—¦ J) вҠ— i | Оҫ вҲҲ T вҲ— M }
= complex-antilinear cotangent vectors ,

and ПҖ 1,0 , ПҖ 0,1 are the two natural projections
ПҖ 1,0 : T вҲ— M вҠ— C вҲ’вҶ’ T 1,0
О· вҲ’вҶ’ О· 1,0 := 1 (О· вҲ’ iО· в—¦ J) ;
2

ПҖ 0,1 : T вҲ— M вҠ— C вҲ’вҶ’ T 0,1
1
О· вҲ’вҶ’ О· 0,1 := 2 (О· + iО· в—¦ J) .

Forms of Type ( , m)
14.2
For an almost complex manifold (M, J), let

в„¦k (M ; C) := sections of Оӣk (T вҲ— M вҠ— C)
= complex-valued k-forms on M, where

Оӣk (T вҲ— M вҠ— C) := Оӣk (T 1,0 вҠ• T 0,1 )
= вҠ• +m=k (Оӣ T 1,0 ) вҲ§ (Оӣm T 0,1 )
,m (deп¬Ғnition)
Оӣ
,m
=вҠ• +m=k Оӣ .

In particular, Оӣ1,0 = T 1,0 and Оӣ0,1 = T 0,1 .

Deп¬Ғnition 14.1 The diп¬Җerential forms of type ( , m) on (M, J) are the
sections of Оӣ ,m :
в„¦ ,m := sections of Оӣ ,m .

Then
в„¦k (M ; C) = вҠ• ,m
+m=k в„¦ .
Let ПҖ ,m : Оӣk (T вҲ— M вҠ— C) вҶ’ Оӣ ,m be the projection map, where + m = k. The
usual exterior derivative d composed with these projections induces diп¬Җerential
ВҜ
operators вҲ‚ and вҲ‚ on forms of type ( , m):
+1,m +1,m
,m
в—¦d :в„¦ (M ) вҲ’вҶ’ в„¦
вҲ‚ := ПҖ (M )
ВҜ ,m+1 ,m ,m+1
в—¦d :в„¦ (M ) вҲ’вҶ’ в„¦
вҲ‚ := ПҖ (M ) .
80 14 DOLBEAULT THEORY

,m
(M ), with k = + m, then dОІ вҲҲ в„¦k+1 (M ; C):
If ОІ вҲҲ в„¦

ВҜ
ПҖ r,s dОІ = ПҖ k+1,0 dОІ + В· В· В· + вҲ‚ОІ + вҲ‚ОІ + В· В· В· + ПҖ 0,k+1 dОІ .
dОІ =
r+s=k+1

14.3 J-Holomorphic Functions

Let f : M вҶ’ 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 ).

Deп¬Ғnition 14.2 A function f is (J-)holomorphic at x вҲҲ M if dfp is com-
plex linear, i.e., dfp в—¦ J = i dfp . A function f is (J-)holomorphic if it is
holomorphic at all p вҲҲ M .

Exercise. Show that

1,0 0,1
dfp в—¦ J = i dfp вҮҗвҮ’ dfp вҲҲ Tp вҮҗвҮ’ ПҖp dfp = 0 .

в™¦

Deп¬Ғnition 14.3 A function f is (J-)anti-holomorphic at p вҲҲ M if dfp is
complex antilinear, i.e., dfp в—¦ J = вҲ’i dfp .

Exercise.
0,1 1,0
dfp в—¦ J = вҲ’i dfp вҮҗвҮ’ dfp вҲҲ Tp вҮҗвҮ’ ПҖp dfp = 0
ВҜ 0,1 ВҜ
1,0
вҮҗвҮ’ d fp вҲҲ Tp вҮҗвҮ’ ПҖp dfp = 0
ВҜ
вҮҗвҮ’ f is holomorphic at p вҲҲ M .

в™¦

ВҜ
Deп¬Ғnition 14.4 On functions, d = вҲ‚ + вҲ‚, where

ВҜ
вҲ‚ := ПҖ 1,0 в—¦ d вҲ‚ := ПҖ 0,1 в—¦ d .
and

Then
ВҜ
вҮҗвҮ’
f is holomorphic вҲ‚f = 0 ,
вҮҗвҮ’
f is anti-holomorphic вҲ‚f = 0 .

вҖ“ What about higher diп¬Җerential forms?
81
14.4 Dolbeault Cohomology

14.4 Dolbeault Cohomology
ВҜ
Suppose that d = вҲ‚ + вҲ‚, i.e.,
ВҜ ,m
вҲҖОІ вҲҲ в„¦
dОІ = вҲ‚ОІ + вҲ‚ОІ , .
+1,m ,m+1
вҲҲв„¦ вҲҲв„¦

,m
Then, for any form ОІ вҲҲ в„¦ ,
ВҜ ВҜ ВҜ
0 = d2 ОІ = вҲ‚ 2 ОІ + вҲ‚ вҲ‚ОІ + вҲ‚вҲ‚ОІ + вҲ‚ 2 ОІ ,
+2,m +1,m+1 ,m+2
вҲҲв„¦ вҲҲв„¦ вҲҲв„¦

which implies пЈ±2
ВҜ
пЈІ вҲ‚ =0
ВҜВҜ
вҲ‚ вҲ‚ + вҲ‚вҲ‚ = 0
пЈі2
вҲ‚ =0
ВҜ
Since вҲ‚ 2 = 0, the chain
ВҜ ВҜ ВҜ
вҲ‚ вҲ‚ вҲ‚
,0 ,1 ,2
0 вҲ’вҶ’ в„¦ вҲ’вҶ’ в„¦ вҲ’вҶ’ в„¦ вҲ’вҶ’ В· В· В·

is a diп¬Җerential complex; its cohomology groups
ВҜ
ker вҲ‚ : в„¦ ,m вҲ’вҶ’ в„¦ ,m+1
,m
HDolbeault (M ) := ВҜ
im вҲ‚ : в„¦ ,mвҲ’1 вҲ’вҶ’ в„¦ ,m
are called the Dolbeault cohomology groups.

в„¦0,0
В  d
вҲ‚В  dвҲ‚ВҜ
В  d
В  d
В
d
в„¦1,0 в„¦0,1
В  d В  d
вҲ‚В  dвҲ‚ вҲ‚В  dвҲ‚
ВҜ ВҜ
В  d В  d
В  d В  d
В
d В
d
в„¦2,0 в„¦1,1 в„¦0,2

. . .
. . .
. . .

ВҜ
вҖ“ When is d = вҲ‚ + вҲ‚? 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] вҲ’ J[v, Jw] вҲ’ J[Jv, w] вҲ’ [v, w] ,
where v and w are vector п¬Ғelds on M , [В·, В·] is the usual bracket
[v, w] В· f := v В· (w В· f ) вҲ’ w В· (v В· f ) , for f вҲҲ C вҲһ (M ) ,
and v В· f = df (v).

(a) Check that, if the map v вҶ’ [v, w] is complex linear (in the sense that
it commutes with J), then N вүЎ 0.
(b) Show that N is actually a tensor, that is: N (v, w) at x вҲҲ M depends
only on the values vx , wx вҲҲ Tx M and not really on the vector п¬Ғelds
v and w.
(c) Compute N (v, Jv). Deduce that, if M is a surface, then N вүЎ 0.

A theorem of Newlander and Nirenberg  states that an almost complex
manifold (M, J) is a complex (analytic) manifold if and only if N вүЎ 0.
Combining (c) with the fact that any orientable surface is symplectic,
we conclude that any orientable surface is a complex manifold, a result
2. Let N be as above. For any map f : R2n вҶ’ C and any vector п¬Ғeld v on
R2n , we have v В· f = v В· (f1 + if2 ) = v В· f1 + i v В· f2 , so that f вҶ’ v В· 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 в—¦ J = i df .
Show that df (N (v, w)) = 0 for all vector п¬Ғelds 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
вҲ—

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 вҶ’ M such that df в—¦ i = J в—¦ df . J-holomorphic curves, also known as
pseudo-holomorphic curves, provide a main tool in symplectic topol-
ogy, as п¬Ғrst realized by Gromov .

82
Part VI
KВЁhler Manifolds
a
KВЁhler geometry lies at the intersection of complex, riemannian and symplec-
a
tic geometries, and plays a central role in all of these п¬Ғelds. We will start by
reviewing complex manifolds. After describing the local normal form for KВЁh- a
ler manifolds (Lecture 16), we conclude with a summary of Hodge theory for
compact KВЁhler manifolds (Lecture 17).
a

15 Complex Manifolds

15.1 Complex Charts

Deп¬Ғnition 15.1 A complex manifold of (complex) dimension n is a set M
with a complete complex atlas

A = {(UО± , VО± , П•О± ) , О± вҲҲ index set I}

where M = вҲӘО± UО± , the VО± вҖ™s are open subsets of Cn , and the maps П•О± : UО± вҶ’
VО± are such that the transition maps ПҲО±ОІ are biholomorphic as maps on open
subsets of Cn :
В  d
П•О± В  d П•ОІ
В  d
В  d
В
d
вҲ’1
ПҲО±ОІ = П•ОІ в—¦ П•О± E
VО±ОІ VОІО±
where VО±ОІ = П•О± (UО± вҲ© UОІ ) вҠҶ Cn and VОІО± = П•ОІ (UО± вҲ© UОІ ) вҠҶ Cn . ПҲО±ОІ being
вҲ’1
biholomorphic means that ПҲО±ОІ is a bijection and that ПҲО±ОІ and ПҲО±ОІ are both
holomorphic.

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

Proof.
1) Local deп¬Ғnition of J:
Let (U, V, П• : U вҶ’ V) be a complex chart for a complex manifold M with
П• = (z1 , . . . , zn ) written in components relative to complex coordinates
zj = xj + iyj . At p вҲҲ U

вҲ‚ вҲ‚
Tp M = R-span of , : j = 1, . . . , n .
вҲ‚xj вҲ‚yj
p p

83
84 15 COMPLEX MANIFOLDS

Deп¬Ғne J over U by

вҲ‚ вҲ‚
Jp =
вҲ‚xj вҲ‚yj
p p
j = 1, . . . , n .
вҲ‚ вҲ‚
=вҲ’
Jp
вҲ‚yj вҲ‚xj
p p

2) This J is well-deп¬Ғned globally:
If (U, V, П•) and (U , V , П• ) are two charts, we need to show that J = J
on their overlap.
On U вҲ© U , ПҲ в—¦ П• = П• . If zj = xj + iyj and wj = uj + ivj are coordinates
on U and U , respectively, so that П• and П• can be written in components
П• = (z1 , . . . , zn ), П• = (w1 , . . . , wn ), then ПҲ(z1 , . . . , zn ) = (w1 , . . . , wn ).
Taking the derivative of a composition
пЈ±
пЈҙвҲ‚ вҲ‚uj вҲ‚ вҲ‚vj вҲ‚
пЈҙ = +
пЈҙ вҲ‚x
пЈІ вҲ‚xk вҲ‚uj вҲ‚xk вҲ‚vj
k j
пЈҙвҲ‚ вҲ‚uj вҲ‚ вҲ‚vj вҲ‚
пЈҙ = +
пЈҙ
пЈі вҲ‚yk вҲ‚yk вҲ‚uj вҲ‚yk вҲ‚vj
j

Since ПҲ is biholomorphic, each component of ПҲ satisп¬Ғes the Cauchy-
Riemann equations:
пЈ±
пЈҙ вҲ‚uj = вҲ‚vj
пЈҙ
пЈҙ вҲ‚x
пЈІ вҲ‚yk
k
j, k = 1, . . . , n .
пЈҙ вҲ‚u
пЈҙ вҲ‚vj
пЈҙ j
пЈі =вҲ’
вҲ‚yk вҲ‚xk
These equations imply
вҲ‚uj вҲ‚ вҲ‚vj вҲ‚ вҲ‚uj вҲ‚ вҲ‚vj вҲ‚
J + = +
вҲ‚xk вҲ‚uj вҲ‚xk вҲ‚vj вҲ‚yk вҲ‚uj вҲ‚yk вҲ‚vj
j j
пЈ« пЈ¶
пЈ¬ пЈ·
пЈ¬ вҲ‚u вҲ‚ вҲ‚vj вҲ‚ пЈ·
пЈ¬j пЈ·
вҲ’
пЈ¬ пЈ·
пЈ¬ вҲ‚xk вҲ‚vj вҲ‚xj вҲ‚uj пЈ·
пЈӯ пЈё
j
вҲ‚vj вҲ‚u
вҲ’ вҲ‚y j
вҲ‚yk
k

which matches the equation
вҲ‚ вҲ‚
J = .
вҲ‚xk вҲ‚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 в„¦k (M ; C) = вҠ• +m=k в„¦ ,m look like? ([22,
48, 65, 107] are good references for this material.)
Let U вҠҶ M be a coordinate neighborhood with complex coordinates z1 , . . . , zn ,
zj = xj + iyj , and real coordinates x1 , y1 , . . . , xn , yn . At p вҲҲ U,

вҲ‚ вҲ‚
Tp M = R-span ,
вҲ‚xj вҲ‚yj
p p
вҲ‚ вҲ‚
Tp M вҠ— C = C-span ,
вҲ‚xj вҲ‚yj
p p

1 вҲ‚ вҲ‚ 1 вҲ‚ вҲ‚
вҲ’i вҠ•
= C-span +i
C-span
2 вҲ‚xj вҲ‚yj 2 вҲ‚xj вҲ‚yj
p p p p

T1,0 = (+i)-eigenspace of J T0,1 = (вҲ’i)-eigenspace of J
вҲ‚ вҲ‚ вҲ‚ вҲ‚ вҲ‚ вҲ‚ вҲ‚ вҲ‚
вҲ’ i вҲ‚yj вҲ’ i вҲ‚yj = вҲ’i
J =i J + i вҲ‚yj + i вҲ‚yj
вҲ‚xj вҲ‚xj вҲ‚xj вҲ‚xj

This can be written more concisely using:

Deп¬Ғnition 15.3
вҲ‚ 1 вҲ‚ вҲ‚ вҲ‚ 1 вҲ‚ вҲ‚
вҲ’i
:= and := +i .
вҲ‚zj 2 вҲ‚xj вҲ‚yj вҲ‚ zj
ВҜ 2 вҲ‚xj вҲ‚yj

Hence,

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

Similarly,

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

= C-span{dxj + idyj : j = 1, . . . , n} вҠ• C-span{dxj вҲ’ idyj : j = 1, . . . , n}
T 1,0 T 0,1
(dxj + idyj ) в—¦ J = i(dxj + idyj ) (dxj вҲ’ idyj ) в—¦ J = вҲ’i(dxj вҲ’ idyj )

Putting
dВҜj = dxj вҲ’ idyj ,
dzj = dxj + idyj and z
we obtain

T 1,0 = C-span{dzj : j = 1, . . . , n} , T 0,1 = C-span{dВҜj : j = 1, . . . , n} .
z
86 15 COMPLEX MANIFOLDS

On the coordinate neighborhood U,

| bj вҲҲ C вҲһ (U; C)
(1, 0)-forms = j bj dzj

| bj вҲҲ C вҲһ (U; C)
(0, 1)-forms = j bj dВҜj
z
bj1 ,j2 dzj1 вҲ§ dzj2 | bj1 ,j2 вҲҲ C вҲһ (U; C)
(2, 0)-forms = j1 <j2

bj1 ,j2 dzj1 вҲ§ dВҜj2 | bj1 ,j2 вҲҲ C вҲһ (U; C)
(1, 1)-forms = z
j1 ,j2

bj1 ,j2 dВҜj1 вҲ§ dВҜj2 | bj1 ,j2 вҲҲ C вҲһ (U; C)
(0, 2)-forms = z z
j1 <j2

If we use multi-index notation:

1 вү¤ j1 < . . . < jm вү¤ n
J = (j1 , . . . , jm )
|J| = m
dzL = dzj1 вҲ§ dzj2 вҲ§ . . . вҲ§ dzjm

then
пЈ± пЈј
пЈІ пЈҪ
bJ,K dzJ вҲ§ dВҜK | bJ,K вҲҲ C вҲһ (U; C)
,m
в„¦ = ( , m)-forms = z .
пЈі пЈҫ
|J|= ,|K|=m

15.3 Diп¬Җerentials

On a coordinate neighborhood U, a form ОІ вҲҲ в„¦k (M ; C) may be written as

with aJ,K вҲҲ C вҲһ (U; C) .
aJ,K dxJ вҲ§ dyK ,
ОІ=
|J|+|K|=k

We would like to know whether the following equality holds:

?
ВҜ ВҜ
(вҲ‚aJ,K + вҲ‚aJ,K )dxJ вҲ§ dyK = (вҲ‚ + вҲ‚) aJ,K dxJ вҲ§ dyK .
dОІ =

If we use the identities

1
dxj + idyj = dzj dxj = 2 (dzj + dВҜj )
z
вҮҗвҮ’ 1
dxj вҲ’ idyj = dВҜj dyj = 2i (dzj вҲ’ dВҜj )
z z
87
15.3 Diп¬Җerentials

after substituting and reshuп¬„ing, we obtain

bJ,K dzJ вҲ§ dВҜK
ОІ = z
|J|+|K|=k

пЈ« пЈ¶
пЈӯ bJ,K dzJ вҲ§ dВҜK пЈё ,
= z
+m=k |J|= ,|K|=m

,m
вҲҲв„¦
пЈ« пЈ¶
пЈӯ dbJ,K вҲ§ dzJ вҲ§ dВҜK пЈё
dОІ = z
+m=k |J|= ,|K|=m

ВҜ
вҲ‚bJ,K + вҲ‚bJ,K вҲ§ dzJ вҲ§ dВҜK
= z
+m=k |J|= ,|K|=m
ВҜ
(because d = вҲ‚ + вҲ‚ on functions)
пЈ« пЈ¶
пЈ¬ пЈ·
пЈ¬ пЈ·
пЈ¬ пЈ·
ВҜ
вҲ‚bJ,K вҲ§ dzJ вҲ§ dВҜK + вҲ§ dzJ вҲ§ dВҜK пЈ·
= z вҲ‚bJ,K z
пЈ¬
пЈ¬ пЈ·
+m=k пЈӯ|J|= пЈё
|J|= ,|K|=m
,|K|=m

+1,m ,m+1
вҲҲв„¦ вҲҲв„¦

ВҜ
= вҲ‚ОІ + вҲ‚ОІ .

ВҜ
Therefore, d = вҲ‚ + вҲ‚ on forms of any degree for a complex manifold.
ВҜ
Conclusion. If M is a complex manifold, then d = вҲ‚ + вҲ‚. (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 вҲҲ C вҲһ (U; C), in terms of z and z , we obtain the following formu-
ВҜ
las:
вҲ‚b вҲ‚b
db = dxj + dyj
вҲ‚xj вҲ‚yj
j
1 вҲ‚b вҲ‚b 1 вҲ‚b вҲ‚b
вҲ’i (dxj вҲ’ idyj )
= (dxj + idyj ) + +i
2 вҲ‚xj вҲ‚yj 2 вҲ‚xj вҲ‚yj
j
вҲ‚b вҲ‚b
= dzj + dВҜj
z .
вҲ‚zj вҲ‚ zj
ВҜ
j

Hence:
вҲ‚b = ПҖ 1,0 db = вҲ‚b
j вҲ‚zj dzj
ВҜ
вҲ‚b = ПҖ 0,1 db = вҲ‚b
j вҲ‚ zj dВҜj
ВҜz
88 15 COMPLEX MANIFOLDS

в™¦
,m
In the case where ОІ вҲҲ в„¦ , we have
ВҜ
dОІ = вҲ‚ОІ + вҲ‚ОІ = ( + 1, m)-form + ( , m + 1)-form
0 = d2 ОІ = ( + 2, m)-form + ( + 1, m + 1)-form + ( , m + 2)-form

ВҜВҜ ВҜ
вҲ‚ 2 ОІ + (вҲ‚ вҲ‚ + вҲ‚вҲ‚)ОІ + вҲ‚ 2 ОІ .
=
0 0
0

ВҜ
Hence, вҲ‚ 2 = 0.
The Dolbeault theorem states that for complex manifolds
,m
HDolbeault(M ) = H m (M ; O(в„¦( ,0)
)) ,

where O(в„¦( ,0)
) is the sheaf of forms of type ( , 0) over M .

Theorem 15.4 (Newlander-Nirenberg, 1957 )
Let (M, J) be an almost complex manifold. Let N be the Nijenhuis tensor
(deп¬Ғned in Homework 10). Then:

вҮҗвҮ’
M is a complex manifold J is integrable
вҮҗвҮ’ N вүЎ0
ВҜ
вҮҗвҮ’ d=вҲ‚+вҲ‚
ВҜ
вҲ‚2 = 0
вҮҗвҮ’
ПҖ 2,0 d|в„¦0,1 = 0 .
вҮҗвҮ’

For the proof of this theorem, besides the original reference, see also [22, 30,
ВҜ
48, 65, 107]. Naturally most almost complex manifolds have d = вҲ‚ + вҲ‚.
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 identiп¬Ғcations (z0 , . . . , zn ) вҲј
(О»z0 , . . . , О»zn ) for all О» вҲҲ 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 О».)
Let Ui be the subset of CPn consisting of all points p = [z0 , . . . , zn ] for which
zi = 0. Let П•i : Ui вҶ’ Cn be the map
ziвҲ’1 zi+1
z0 zn
П•i ([z0 , . . . , zn ]) = zi , . . . , zi , zi , . . . , zi .

1. Show that the collection

{(Ui , Cn , П•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 , П•0 ) and
Hint:
n , П• ). Show that the transition diagram has the form
(U1 , C 1

В  dП•
П•0 В  d1
В  d
В
В‚
E V1,0
П•0,1
V0,1
where V0,1 = V1,0 = {(z1 , . . . , zn ) вҲҲ Cn | z1 = 0} and
z2
, . . . , zn )
1
П•0,1 (z1 , . . . , zn ) = ( z , .
z1 z1
1

2. Show that the 1-dimensional complex manifold CP1 is diп¬Җeomorphic, as a
real 2-dimensional manifold, to S 2 .

Stereographic projection.
Hint:

89
16 KВЁhler Forms
a

16.1 KВЁhler Forms
a

Deп¬Ғnition 16.1 A KВЁhler manifold is a symplectic manifold (M, Пү) equipped
a
with an integrable compatible almost complex structure. The symplectic form Пү
is then called a KВЁhler form.
a

It follows immediately from the previous deп¬Ғnition that

(M, Пү) is KВЁhler =вҮ’ M is a complex manifold
a

в„¦k (M ; C) = вҠ• ,m
+m=k в„¦
=вҮ’ ВҜ
d=вҲ‚+вҲ‚

where
+1,m +1,m
,m
в—¦d:в„¦ вҶ’в„¦
вҲ‚=ПҖ
ВҜ ,m+1 ,m ,m+1
в—¦d:в„¦ вҶ’в„¦
вҲ‚=ПҖ .
On a complex chart (U, z1 , . . . , zn ), n = dimC M ,
пЈ± пЈј
пЈІ пЈҪ
вҲһ
,m
bJK dzJ вҲ§ dВҜK | bJK вҲҲ C (U; C) ,
в„¦ = z
пЈі пЈҫ
|J|= ,|K|=m

where
= dzj1 вҲ§ . . . вҲ§ dzj ,
J = (j1 , . . . , j ) , j1 < . . . < j , dzJ
= dВҜk1 вҲ§ . . . вҲ§ dВҜkm .
K = (k1 , . . . , km ) , k1 < . . . < k m , dВҜK
z z z

On the other hand,

(M, Пү) is KВЁhler =вҮ’ Пү is a symplectic form .
a

вҖ“ Where does Пү п¬Ғt with respect to the above decomposition?
A KВЁhler form Пү 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 KВЁhler Forms
a

1. в„¦2 (M ; C) = в„¦2,0 вҠ• в„¦1,1 вҠ• в„¦0,2 .
On a local complex chart (U, z1 , . . . , zn ),

ajk dzj вҲ§ dzk + bjk dzj вҲ§ dВҜk + cjk dВҜj вҲ§ dВҜk
Пү= z z z

for some ajk , bjk , cjk вҲҲ C вҲһ (U; C).
2. J is a symplectomorphism, that is, J вҲ— Пү = Пү where (J вҲ— Пү)(u, v) := Пү(Ju, Jv).

J вҲ— dzj = dzj в—¦ J = idzj
J вҲ— dВҜj = dВҜj в—¦ J = вҲ’idВҜj
z z z
вҲ’1
1
вҲ’1

J вҲ—Пү bjk dzj вҲ§ dВҜk + (вҲ’i)2
(i В· i) ajk dzj вҲ§ dzk + i(вҲ’i) cjk d zj вҲ§ dВҜk
= z ВҜ z

J вҲ—Пү Пү вҲҲ в„¦1,1 .
вҮҗвҮ’ вҮҗвҮ’
=Пү ajk = 0 = cjk , all j, k

вҲ‚Пү = 0 Пү is вҲ‚-closed
ВҜ
3. 0 = dПү = вҲ‚Пү + вҲ‚Пү =вҮ’ ВҜ ВҜ
вҲ‚Пү = 0 Пү is вҲ‚-closed
(2,1)вҲ’form (1,2)вҲ’form

Hence, Пү deп¬Ғnes a Dolbeault (1, 1) cohomology class,
1,1
[Пү] вҲҲ HDolbeault (M ) .
i
Putting bjk = 2 hjk ,
n
i
hjk вҲҲ C вҲһ (U; C).
hjk dzj вҲ§ dВҜk ,
Пү= z
2
j,k=1

4. Пү real-valued вҮҗвҮ’ Пү = Пү.
i i i
Пү=вҲ’ hjk dВҜj вҲ§ dzk = hjk dzk вҲ§ dВҜj = hkj dzj вҲ§ dВҜk
z z z
2 2 2
вҮҗвҮ’ hjk = hkj ,
Пү real
i.e., at every point p вҲҲ U, the n Г— n matrix (hjk (p)) is hermitian.
5. nondegeneracy: Пү n = Пү вҲ§ . . . вҲ§ Пү = 0.
n

Exercise. Check that
n
i
n
det(hjk ) dz1 вҲ§ dВҜ1 вҲ§ . . . вҲ§ dzn вҲ§ dВҜn .
Пү = n! z z
2
в™¦

Now
вҮҗвҮ’ detC (hjk ) = 0 ,
Пү nondegenerate
i.e., at every p вҲҲ M , (hjk (p)) is a nonsingular matrix.
ВЁ
92 16 KAHLER FORMS

2. Again the positivity condition: Пү(v, Jv) > 0, вҲҖv = 0.

в™¦
Exercise. Show that (hjk (p)) is positive-deп¬Ғnite.

Пү positive вҮҗвҮ’ (hjk ) 0,

i.e., at each p вҲҲ U, (hjk (p)) is positive-deп¬Ғnite.

ВҜ
Conclusion. KВЁhler forms are вҲ‚- and вҲ‚-closed (1, 1)-forms, which are given on
a
a local chart (U, z1 , . . . , zn ) by
n
i
hjk dzj вҲ§ dВҜk
Пү= z
2
j,k=1

where, at every point p вҲҲ U, (hjk (p)) is a positive-deп¬Ғnite hermitian matrix.

16.2 An Application
Theorem 16.2 (Banyaga) Let M be a compact complex manifold. Let Пү0
2
and Пү1 be KВЁhler forms on M . If [Пү0 ] = [Пү1 ] вҲҲ HdeRham (M ), then (M, Пү0 ) and
a
(M, Пү1 ) are symplectomorphic.

Proof. Any combination Пүt = (1 вҲ’ t)Пү0 + tПү1 is symplectic for 0 вү¤ t вү¤ 1,
because, on a complex chart (U, z1 , . . . , zn ), where n = dimC M , we have
i
h0 dzj вҲ§ dВҜk
Пү0 = z
jk
2

i
h1 dzj вҲ§ dВҜk
Пү1 = z
jk
2

i
where ht = (1 вҲ’ t)h0 + th1 .
ht dzj вҲ§ dВҜk ,
Пүt = z
jk jk jk jk
2

(h0 ) 0 , (h1 ) 0 =вҮ’ (ht ) 0.
jk jk jk

Apply the Moser theorem (Theorem 7.2).

16.3 Recipe to Obtain KВЁhler Forms
a
Deп¬Ғnition 16.3 Let M be a complex manifold. A function ПҒ вҲҲ C вҲһ (M ; R) is
strictly plurisubharmonic (s.p.s.h.) if, on each local complex chart (U, z 1 , . . . , zn ),
вҲ‚2ПҒ
is positive-deп¬Ғnite at all p вҲҲ U.
where n = dimC M , the matrix вҲ‚zj вҲ‚ zk (p)
ВҜ

Proposition 16.4 Let M be a complex manifold and let ПҒ вҲҲ C вҲһ (M ; R) be
s.p.s.h.. Then
iВҜ
Пү = вҲ‚ вҲ‚ПҒ is KВЁhler .
a
2
93
16.3 Recipe to Obtain KВЁhler Forms
a

A function ПҒ as in the previous proposition is called a (global) KВЁhler
a
potential.
Proof. Simply observe that:
пЈ± 0
пЈҙ
пЈҙ
пЈҙ
пЈҙ вҲ‚Пү ВҜ
вҲ‚ 2 вҲ‚ПҒ = 0
i
пЈІ = 2

пЈҙВҜ
пЈҙ вҲ‚Пү ВҜВҜ ВҜ
вҲ‚вҲ‚ вҲ‚ПҒ = вҲ’ 2 вҲ‚ вҲ‚ 2 ПҒ = 0
i i
пЈҙ =
пЈҙ
пЈі 2
ВҜ 0
вҲ’вҲ‚ вҲ‚

ВҜ
dПү = вҲ‚Пү + вҲ‚Пү = 0 =вҮ’ Пү is closed .

iВҜ iВҜ
Пү = вҲ’ 2 вҲ‚вҲ‚ПҒ = 2 вҲ‚ вҲ‚ПҒ = Пү =вҮ’ Пү is real .

Пү вҲҲ в„¦1,1 =вҮ’ J вҲ— Пү = Пү =вҮ’ Пү(В·, JВ·) is symmetric .

Exercise. Show that, for f вҲҲ C вҲһ (U; C),

вҲ‚f вҲ‚f
ВҜ
вҲ‚f = dzj and вҲ‚f = dВҜj .
z
вҲ‚zj вҲ‚ zj
ВҜ

Since the right-hand sides are in в„¦1,0 and в„¦0,1 , respectively, it suп¬ғces to show
в™¦
that the sum of the two expressions is df .

вҲ‚2ПҒ
iВҜ i вҲ‚ вҲ‚ПҒ i
dzj вҲ§ dВҜk = dzj вҲ§ dВҜk .
Пү = вҲ‚ вҲ‚ПҒ = z z
2 2 вҲ‚zj вҲ‚ zk
ВҜ 2 вҲ‚zj вҲ‚ zk
ВҜ
hjk

ПҒ is s.p.s.h =вҮ’ (hjk ) 0 =вҮ’ Пү(В·, JВ·) is positive .
In particular, Пү 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 =
ПҒ(x1 , y1 , . . . , xn , yn ) = zj z j .
ВҜ
j
j=1

Then
вҲ‚ вҲ‚ПҒ вҲ‚
= zk = Оҙjk ,
вҲ‚zj вҲ‚ zk
ВҜ вҲ‚zj
so
вҲ‚2ПҒ
(hjk ) = = (Оҙjk ) = Id 0 =вҮ’ ПҒ is s.p.s.h. .
вҲ‚zj вҲ‚ zk
ВҜ
ВЁ
94 16 KAHLER FORMS

The corresponding KВЁhler form
a
iВҜ i
Оҙjk dzj вҲ§ dВҜk
Пү = 2 вҲ‚ вҲ‚ПҒ = z
2
j,k

i
dzj вҲ§ dВҜj = dxj вҲ§ dyj
= z is the standard form .
2
j j

в™¦

16.4 Local Canonical Form for KВЁhler Forms
a

There is a local converse to the previous construction of KВЁhler forms.
a

Theorem 16.5 Let Пү be a closed real-valued (1, 1)-form on a complex manifold
M and let p вҲҲ M . Then there exist a neighborhood U of p and ПҒ вҲҲ C вҲһ (U; R)
such that, on U,
iВҜ
Пү = вҲ‚ вҲ‚ПҒ .
2

The function ПҒ is then called a (local) KВЁhler potential.
a
The proof requires holomorphic versions of PoincarВҙвҖ™s lemma, namely, the
e
local triviality of Dolbeault groups:
,m
вҲҖp вҲҲ M вҲғ neighborhood U of p such that HDolbeault (U) = 0 , m > 0 ,

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

Theorem 16.6 Let M be a complex manifold, ПҒ вҲҲ C вҲһ (M ; R) s.p.s.h., X a
complex submanifold, and i : X вҶ’ M the inclusion map. Then iвҲ— ПҒ is s.p.s.h..

Proof. Let dimC M = n and dimC X = n вҲ’ m. For p вҲҲ X, choose a chart
(U, z1 , . . . , zn ) for M centered at p and adapted to X, i.e., X вҲ© U is given by
z1 = . . . = zm = 0. In this chart, iвҲ— ПҒ = ПҒ(0, 0, . . . , 0, zm+1 , . . . , zn ).

вҲ‚2ПҒ
вҲ—
i p is s.p.s.h. вҮҗвҮ’ (0, . . . , 0, zm+1 , . . . , zn ) is positive-deп¬Ғnite ,
вҲ‚zm+j вҲ‚ zm+k
ВҜ

вҲ‚2
which holds since this is a minor of вҲ‚zj вҲ‚ zk (0, . . . , 0, zm+1 , . . . , zn ) .
ВҜ

Corollary 16.7 Any complex submanifold of a KВЁhler manifold is also KВЁhler.
a a

Deп¬Ғnition 16.8 Let (M, Пү) be a KВЁhler manifold, X a complex submanifold,
a
and i : X вҶ’ M the inclusion. Then (X, iвҲ— Пү) is called a KВЁhler submanifold.
a
95
16.4 Local Canonical Form for KВЁhler Forms
a

i
Example. Complex vector space (Cn , Пү) where Пү = dzj вҲ§ dВҜj is KВЁhler.
z a
2
Every complex submanifold of Cn is KВЁhler. в™¦
a

Example. The complex projective space is

CPn = Cn+1 \{0}/ вҲј

where
(z0 , . . . , zn ) вҲј (О»z0 , . . . , О»zn ) , О» вҲҲ C\{0} .
The Fubini-Study form (see Homework 12) is KВЁhler. Therefore, every non-
a
singular projective variety is a KВЁhler submanifold. Here we mean
a

non-singular = smooth
projective variety = zero locus of a collection
of homogeneous polynomials .

в™¦
Homework 12: The Fubini-Study Structure
The purpose of the following exercises is to describe the natural KВЁhler structure
a
n
on complex projective space, CP .

1. Show that the function on Cn

z вҲ’вҶ’ log(|z|2 + 1)

is strictly plurisubharmonic. Conclude that the 2-form
iВҜ
ПүFS = 2 вҲ‚ вҲ‚ log(|z|2 + 1)

is a KВЁhler form. (It is usually called the Fubini-Study form on Cn .)
a
Hint: A hermitian n Г— n matrix H is positive deп¬Ғnite if and only if v вҲ— Hv > 0
for any v вҲҲ Cn \ {0}, where v вҲ— is the transpose of the vector v . To prove
ВҜ
positive-deп¬Ғniteness, either apply the Cauchy-Schwarz inequality, or use the
following symmetry observation: U(n) acts transitively on S 2nвҲ’1 and ПүFS is
U(n)-invariant, thus it suп¬ғces to show positive-deп¬Ғniteness along one direction.

2. Let U be the open subset of Cn deп¬Ғned by the inequality z1 = 0, and let
П• : U вҶ’ U be the map
1
П•(z1 , . . . , zn ) = z1 (1, z2 , . . . , zn ) .

Show that П• maps U biholomorphically onto U and that

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

3. Notice that, for every point p вҲҲ U, we can write the second term in ( ) as
the sum of a holomorphic and an anti-holomorphic function:

вҲ’ log z1 вҲ’ log z1

on a neighborhood of p. Conclude that
ВҜ ВҜ
вҲ‚ вҲ‚П•вҲ— log(|z|2 + 1) = вҲ‚ вҲ‚ log(|z|2 + 1)

and hence that П•вҲ— ПүFS = ПүFS .
Hint: You need to use the fact that the pullback by a holomorphic map П•вҲ—
ВҜ
commutes with the вҲ‚ and вҲ‚ operators. This is a consequence of П•вҲ— preserving
вҲ— (в„¦p,q ) вҠҶ в„¦p,q , which in turn is implied by П•вҲ— dz = вҲ‚П• вҠҶ в„¦1,0
form type, П• j j
ВҜ
and П•вҲ— dzj = вҲ‚П•j вҠҶ в„¦0,1 , where П•j is the jth component of П• with respect to
local complex coordinates (z1 , . . . , zn ).

4. Recall that CPn is obtained from Cn+1 \ {0} by making the identiп¬Ғca-
tions (z0 , . . . , zn ) вҲј (О»z0 , . . . , О»zn ) for all О» вҲҲ C \ {0}; [z0 , . . . , zn ] is the
equivalence class of (z0 , . . . , zn ).

96
97
HOMEWORK 12

For i = 0, 1, . . . , n, let
Ui = {[z0 , . . . , zn ] вҲҲ CPn |zi = 0}
ziвҲ’1 zi+1
z0 zn
П•i : Ui вҶ’ C n П•i ([z0 , . . . , zn ]) = zi , . . . , zi , zi , . . . , zi .

Homework 11 showed that the collection {(Ui , Cn , П•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 , П•0 ) and
(U1 , Cn , П•1 ) has the form
В  d
В  d П•1
П•0
В  d
В  d
В
d
П•0,1 E V1,0
V0,1
where V0,1 = V1,0 = {(z1 , . . . , zn ) вҲҲ Cn | z1 = 0} and П•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 П• coincides with П•0,1 .
Show that П•вҲ— ПүFS and П•вҲ— ПүFS are identical on the overlap U0 вҲ© U1 .
0 1
More generally, show that the KВЁhler forms П•вҲ— ПүFS вҖңglue togetherвҖқ to
a i
n
deп¬Ғne a KВЁhler 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 ] вҲҲ
CP1 |z0 = 0} is given by the formula
dx вҲ§ dy
ПүFS =
(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 вҲӘ {вҲһ} with respect to ПүFS :
dx вҲ§ dy
ПүFS = .
(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 Пүstd induced by regarding it as the
unit sphere in R3 (Homework 6): in cylindrical polar coordinates (Оё, h) on
S 2 away from its poles (0 вү¤ Оё < 2ПҖ and вҲ’1 вү¤ h вү¤ 1), we have
Пүstd = dОё вҲ§ dh .
Using stereographic projection, show that
1
ПүFS = Пү.
4 std
17 Compact KВЁhler Manifolds
a
17.1 Hodge Theory
Let M be a complex manifold. A KВЁhler form Пү on M is a symplectic form which
a
is compatible with the complex structure. Equivalently, a KВЁhler form Пү is a вҲ‚-
a
ВҜ
and вҲ‚-closed form of type (1, 1) which, on a local chart (U, z1 , . . . , zn ) is given by
n
i
Пү = 2 j,k=1 hjk dzj вҲ§ dВҜk , where, at each x вҲҲ U, (hjk (x)) is a positive-deп¬Ғnite
z
hermitian matrix. The pair (M, Пү) is then called a KВЁhler manifold.
a
Theorem 17.1 (Hodge) On a compact KВЁhler manifold (M, Пү) 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 п¬Ғnite-dimensional.
with H
Hodge identiп¬Ғed the spaces of cohomology classes of forms with spaces of
actual forms, by picking the representative from each class which solves a certain
diп¬Җerential equation, namely the harmonic representative.
(1) The Hodge вҲ—-operator.
Each tangent space V = Tx M has a positive inner product В·, В· , 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 вҲ— : Оӣ(V ) вҶ’ Оӣ(V ) deп¬Ғned by
вҲ—(1) = e1 вҲ§ . . . вҲ§ en
вҲ—(e1 вҲ§ . . . вҲ§ en ) = 1
вҲ—(e1 вҲ§ . . . вҲ§ ek ) = ek+1 вҲ§ . . . вҲ§ en .

We see that вҲ— : Оӣk (V ) вҶ’ ОӣnвҲ’k (V ) and satisп¬Ғes вҲ—вҲ— = (вҲ’1)k(nвҲ’k) .
(2) The codiп¬Җerential and the laplacian are the operators deп¬Ғned by:
= (вҲ’1)n(k+1)+1 вҲ— dвҲ— : в„¦k (M ) вҶ’ в„¦kвҲ’1 (M )
Оҙ
: в„¦k (M ) вҶ’ в„¦k (M ) .
вҲҶ = dОҙ + Оҙd
The operator вҲҶ is also called the Laplace-Beltrami operator.
вҲ‚2
n
Exercise. Check that, on в„¦0 (Rn ) = C вҲһ (Rn ), вҲҶ = вҲ’ в™¦
i=1 вҲ‚x2 .
i

Exercise. Check that вҲҶвҲ— = вҲ—вҲҶ. в™¦

Suppose that M is compact. Deп¬Ғne an inner product on forms by

В·, В· : в„¦k Г— в„¦k вҶ’ R , О± вҲ§ вҲ—ОІ .
О±, ОІ =
M

98
99
17.1 Hodge Theory

Exercise. Check that this is symmetric, positive-deп¬Ғnite and satisп¬Ғes
в™¦
dО±, ОІ = О±, ОҙОІ .

Therefore, Оҙ is often denoted by dвҲ— 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 В·, В· : в„¦k Г— в„¦k вҶ’ R deп¬Ғned by a similar
c
formula, where в„¦k is the space of compactly supported k-forms.) Also, вҲҶ
c

Exercise. Check that вҲҶО±, ОІ = О±, вҲҶОІ , and that вҲҶО±, О± = |dО±|2 +
|ОҙО±|2 вүҘ 0, where | В· | is the norm with respect to this inner product. в™¦

(3) The harmonic k-forms are the elements of Hk := {О± вҲҲ в„¦k | вҲҶО± = 0}.

Note that вҲҶО± = 0 вҮҗвҮ’ dО± = ОҙО± = 0. Since a harmonic form is d-closed,
it deп¬Ғnes 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 п¬Ғnite-dimensional. We also have the
following orthogonal decomposition with respect to В·, В· :

в„¦k Hk вҠ• вҲҶ(в„¦k (M ))
(Hodge decomposition on forms) .
Hk вҠ• dв„¦kвҲ’1 вҠ• Оҙв„¦k+1

The proof involves functional analysis, elliptic diп¬Җerential operators, pseu-
dodiп¬Җerential 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.
ВҜВҜ ВҜВҜ
When M is KВЁhler, the laplacian satisп¬Ғes вҲҶ = 2(вҲ‚ вҲ‚ вҲ— + вҲ‚ вҲ— вҲ‚) (see, for
a
example, ) and preserves the decomposition according to type, вҲҶ :
в„¦ ,m вҶ’ в„¦ ,m . Hence, harmonic forms are also bigraded

Hk = ,m
H .
+m=k

Theorem 17.3 (Hodge) Every Dolbeault cohomology class on a com-
pact KВЁhler manifold (M, Пү) possesses a unique harmonic representative,
a
i.e.,
,m
H ,m HDolbeault(M )
ВЁ
100 17 COMPACT KAHLER MANIFOLDS

,m
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