3.6. HODGE THEORY ON KAHLER MANIFOLDS 113

and similarly δ(mdy) = ’(1/f )‚m/‚y. So one sees that actually δ is

’(1/f ) times the divergence of the vector ¬eld (h, m). Now for the

laplacian ∆ one has

‚f ‚h

1 ‚2h ‚ 2h 1 1 ‚x ‚x

∆(hdx) = ’ ( 2 + 2 )dx + 2 ( f · h)dx + 2 (det )dy,

‚f ‚h

f ‚x ‚y f f ‚y ‚y

so there are terms both with dx and dy. However, for a 1-form of pure

type (0, 1) the result is somewhat astonishing:

√ ‚f ‚h

1 ‚2h ‚ 2h 1 ’1 ‚x ‚x

∆(hdz) = [’ ( 2 + 2 ) + 2 ( f · h) ’ 2 (det ‚f ‚h )]dz.

f ‚x ‚y f f ‚y ‚y

It means that a form of type (1, 0) is sent by ∆ again to a form of type

(1, 0).

Before we state and prove more general theorem, we need to in-

troduce more operators on the space of j-forms on a K¨hler mani-

a

fold (M, J, ω). First, we have the adjoint operator δ for the exterior

di¬erential d = d + d . Now let δ : Ap,q (M ) ’ Ap’1,q (M ) and

δ : Ap,q (M ) ’ Ap,q’1 (M ) be the adjoint operators to d and d re-

spectively, so that δ = δ + δ . Also, the K¨hler form ω gives rise to

a

the (Lefschetz) operator

L : Ap,q (M ) ’ Ap+1,q+1 (M ), L± = ω§±.

Let Λ : Ap,q (M ) ’ Ap’1,q’1 (M ) be the adjoint to L. The operators Λ

and L are purely punctual operators. The main tool in proving that ∆

preserves types of di¬erential forms will be the following

LEMMA 3.6.1 √

’1δ = [Λ, d ].

Proof. Because all the operators involved are punctual, we can do

everything just at one point. Because of the local nature of the result

we may assume that M = Cn and our arbitrary point is the origin.

Due to Proposition 1.8 we may assume that ω = ω0 + ω , where ω0 =

√

’1

’2 i dzi §d¯i and ω has only terms vanishing at least to order 2 at

z

the origin, so that ω ∈ M2 A2 (M ), where M0 is the ideal of functions

0

vanishing at 0. If Λ0 , δ0 , and d0 are the operators corresponding to

114 CHAPTER 3. HODGE THEORY

the ”¬‚at” part ω0 , then Λ ’ Λ0 ∈ M2 (Π), where Π is the set of linear

0

operators acting on the space of di¬erential forms on M . Clearly, d0 =

d , and δ ’ δ0 ∈ M0 (Π). One can see that now

[Λ, d ] ’ [Λ0 , d ] ∈ M0 (Π),

and hence

√ √

[Λ, d ] ’ ’1δ ’ ([Λ0 , d ] ’ ’1δ0 ) ∈ M0 (Π).

Since all the operators involved are punctual and we can do the same

procedure√ any arbitrarily chosen point, it is enough to show that

at

[Λ0 , d ] ’ ’1δ0 = 0.

Let us consider Cn as C • C • · · · • C. It is true that the space of

di¬erential forms can be represented as the completed tensor product

ˆ ˆ

of pre-Hilbert spaces of each summand: A— (Cn ) = A— (C)— · · · —A— (C).

We can decompose L = L1 + · · · + Ln , where Li operates on i-th factor.

Now we have the formula

(’1)deg(±1 )+··· deg(±i’1 ) ±1 § · · · §d ±i § · · · §±n .

d (±1 § · · · §±n ) =

i

We have a similar formula for δ which is obtained by adjunction. Thus

everything comes down to the proof of the formula for M = C with

√

’1

ω = ’ 2 dz§d¯. z

First, we ¬nd the number a such that Λ(dz§d¯) = a. If (., .) is our

z

hermitian pairing, then

(L(1), dz§d¯) = (1, Λ(dz§d¯)) = (1, a) = a.

z z ¯

On the other hand,

√

√

’1

(L(1), dz§d¯) = ’

z (dz§d¯, dz§d¯) = ’2 ’1,

z z

2

√

since (dz, dz) = 2, and hence a = 2 ’1. Due to the fact that δ

decreases the holomorphic degree by 1 and Λ decreases both degrees

√

by 1, it is enough to check the identity [Λ, d ] = ’1δ only on forms

of type (1, 1) and (1, 0). In the former case one has

√ ‚f

‚f

d (f dz) = ’ dz§d¯, and Λd (f dz) = 2 ’1 .

z

‚z ‚z

¨

3.6. HODGE THEORY ON KAHLER MANIFOLDS 115

It remains to use our previous knowledge that δ (f dz) = ’2 ‚f . Now,

‚z

‚f

in the latter case we know that δ (f dz§d¯) = ’2 ‚ z d¯ and we get

z z

¯

√ √ ‚f

[Λ, d ]f dz§d¯ = ’d (2 ’1f ) = ’2 ’1 d¯ = δ (f dz§d¯).

z z z

‚z

¯

As a consequence we get

COROLLARY 3.6.2 (1). The operators δ and d anti-commute.

(2). The operators δ and d anti-commute.

Proof. We prove only part (1), because part (2) has the same demon-

stration. Using the lemma above we have

√ √

δ d = ’ ’1[Λ, d ]d = ’1d Λd ,

and √

d δ = ’ ’1d [Λ, d ] = ’δ d .

Let us introduce the operators ∆ = d δ + δ d , ∆ = d δ + δ d .

The operators ∆ and ∆ are called holomorphic and anti-holomorphic

Laplacians respectively.

THEOREM 3.6.3 For any K¨hler manifold M the operator ∆ pre-

a

serves types of di¬erential forms.

Proof. We have

∆ = dδ + δd = (d + d )(δ + δ ) + (δ + δ )(d + d ) =

= (d δ +δ d )+(d δ +δ d )+(δ d +d δ )+(δ d +d δ ) = ∆ +∆ +0+0,

due to the above Corollary. Clearly, ∆ and ∆ preserve types, and the

statement follows.

COROLLARY 3.6.4 If ± is a harmonic form, then all its (p, q) -

components ±p,q are harmonic too.

116 CHAPTER 3. HODGE THEORY

For a compact K¨hler manifold M this result together with Hodge

a

theorem gives us the following conclusion;

THEOREM 3.6.5 The space Hj (M ) of degree j harmonic forms de-

composes as the direct sum

Hj (M ) = •p+q=j H p,q ,

where H p,q is the space of harmonic forms of type (p, q).

Also we mention the following useful fact:

∆ =∆ .

For this we introduce the notion of a real operator on the space A• (M, C)

of di¬erential forms with complex coe¬cients. This pre-Hilbert space

is the complexi¬cation of the real pre-Hilbert space A• (M ) of di¬eren-

tial forms with real coe¬cients. If P is an operator on complex-valued

¯

di¬erential forms, the complex-conjugate operator is P = iP i, where i

¯

is complex-conjugation on A• (M, C). We say that P is real if P = P ,

or equivalently if P maps A• (M ) to itself. For instance, d is a real

operator, and so is ∆. The complex-conjugate of d is d . Now write

√

∆ = d δ + δ d = ’ ’1(d Λd ’ d d Λ + Λd d ’ d Λd ).

We notice that

d Λd = d Λd , and d d = d d = ’d d .

¯

Thus ∆ is a real operator. But, of course, ∆ = ∆ . Therefore for any

K¨hler manifold M we have

a

∆ = 2∆ = 2∆ = ∆ + ∆ .

We can use the same arguments as in Lemma 1.10 to see that ∆ ± =

0 if and only if both d ± = 0 and δ ± = 0. In this way we obtain

¨

3.6. HODGE THEORY ON KAHLER MANIFOLDS 117

COROLLARY 3.6.6 Let M be a compact K¨hler manifold. For a

a

di¬erential j-form ± on M the following are equivalent:

· ± is harmonic,

· d± = δ± = 0

· d±=δ±=0

· d ±=δ ±=0

· each (p, q) component ±p,q is harmonic

· d ±p,q = δ ±p,q = 0 for each (p, q), etc.

The next question which we face is how to construct harmonic forms

on a compact K¨hler manifold. The explicit construction is di¬cult to

a

¬nd in general, though we have

PROPOSITION 3.6.7 Let ± be a holomorphic j-form on a compact

K¨hler manifold M . Then ± is harmonic (hence closed) and it is not

a

exact unless ± = 0.

Proof. We recall that ± is a holomorphic j-form if locally it can be

written as

±= fi1 ...ij dzi1 § · · · §dzij ,

i1 <···<ij

and fi1 ...ij are holomorphic functions. Let us notice that a di¬erential

form β of type (j, 0) is holomorphic if and only if d β = 0. Thus

d ± = 0, and by dimension reasons δ ± = 0. By the above Corollary,

± is harmonic.

COROLLARY 3.6.8 The space „¦j (M ) ‚ H j (M, C) of global holo-

morphic j - forms on M is ¬nite-dimensional.

It is sometimes useful to know that if M is a compact riemannian

manifold then the space of di¬erential forms on M admits the direct

sum decomposition A— (M ) = H • (∆A— (M )), where H is the space of

harmonic forms on M . In general under some mild assumptions this

is a general property of self-adjoint operators. (∆ is an example of

self-adjoint operator.)

Let us give more relations between the di¬erential operators that

have been introduced before. Using duality and complex conjugation

we see that

√ √

[L, δ ] = ’[Λ, d ]— = ’ ’1d , and [L, δ ] = ’ ’1d .

118 CHAPTER 3. HODGE THEORY

LEMMA 3.6.9 The operators L and ∆ commute. Therefore if ± is a

harmonic form then ω§± is harmonic as well.

Proof. Let us prove the equivalent assirtion that [Λ, ∆] = 0. We saw

that [L, d ] = 0, hence [Λ, δ ] = 0 too. We also know that [Λ, δ ] = 0.

Taking into account that ∆ = ∆ = d δ + δ d we have

2

√

[Λ, ∆ ] = [Λ, d ]δ + δ [Λ, d ] = ’ ’1(δ δ + δ δ ) = 0,

√

because [Λ, d ] = ’ ’1δ as follows from Lemma(?.?).

Example. We immediately obtain that on CPn the forms 1, ω, ω§ω, ...

give a basis of the harmonic forms.

LEMMA 3.6.10 If M is a compact riemannian manifold and G a

connected Lie group of isometries of M , then every harmonic form on

M is G-invariant.

Proof. We know that there is an isomorphism between the space Hj

of harmonic j-forms on M and H j (M, C), which has to commute with

the action of G. It remains to notice that in view of the homotopy

invariance principle, G acts trivially on H j (M, C).

Example. Let “ be a lattice in Cn - a discrete cocompact subgroup.

Let us consider the complex torus M = Cn /“. The above Lemma shows

that if ± is a harmonic form on M then it has to be translation invariant,

hence it has to have constant coe¬cients. Let “— = Hom(“, Z), and

then H j (M, Z) = §j “— . It implies that H j (M, C) = §j H 1 (M, C). We

have the decomposition H 1 (M, C) = H 1,0 • H 0,1 , thus

H j (M, C) = •p+q=j (§p H 1,0 ) — (§q H 0,1 ).

We can ¬nd an explicit basis in the space of harmonic forms Hj on M ,

namely dzI §d¯J , where #(I) = p, and #(J) = q. In this particular

z