¯ 2 ‚xi ‚yi

Actually, ‚zi and ‚‚ i are vector ¬elds ”with complex coe¬cients”, mean-

‚

z

¯

ing that they are sections of complexi¬ed tangent bundle T M — C =

Th M • Th M , which we represent as the direct sum of the holomor-

phic and anti-holomorphic tangent bundles to M . In the complexi¬ed

tangent bundle the√ operator J is diagonalizable and Th M corresponds

to√ eigenvalue + ’1 and Th M is an eigenspace with an eigenvalue

its

’ ’1. Indeed,

√ √

‚ 1 ‚ ‚ ‚

J = (J ’ ’1J ) = ’1 .

‚zi 2 ‚xi ‚yi ‚zi

√

Another property of Th M is that it is the range of Id ’ ’1J. This

√ √ √

follows from the equality J(v ’ ’1Jv) = ’1(v ’ ’1Jv).

¨

3.2. KAHLER MANIFOLDS 97

If we look at the dual picture then we will see that T — M — C =

—

—

Th M • Th M allows us to view forms of type (1, 0) as sections of the

—

holomorphic cotangent bundle Th M and forms of type (0, 1) as sections

—

of Th M . Now if ω is a real form of type (1, 1) then it satis¬es ω(v, w) =

0 for v, w ∈ “(Th M ). We write

√ √

v = ξ ’ ’1Jξ and w = · ’ ’1J·,

with ξ and · sections of real tangent bundle. Expanding the equality

√

0 = ω(v, w) = ω(ξ, ·) ’ ω(Jξ, J·) ’ ’1[ω(ξ, J·) + ω(Jξ, ·)]

we see that ω(ξ, ·) = ω(Jξ, J·) and hence ω = Im(H) for a hermitian

form H. Thus a hermitian structure H on a manifold is really equivalent

to the fact that ω = Im(H) is of type (1, 1).

COROLLARY 3.2.4 A complex manifold (M, J) is K¨hler if anda

only if there exists a real non-degenerate closed two-form ω on M of

type (1, 1) satisfying ω(Jv, Jw) = ω(v, w).

If on Cn the riemannian metric g comes from a hermitian form, we have

dzi — d¯j + d¯j — dzi

z z

g= aij dzi d¯j , where dzi d¯j =

z z ,

2

1¤i,j¤n

which is the usual mapping of the symmetric algebra into the tensor

algebra. The reality condition here is

‚‚ aij

aij = aji , g(

¯ , )= .

‚zi ‚ zj

¯ 2

Then we have

√

‚ ‚ ‚ ‚ ‚ ‚

H( , ) = g( , ) + ’1ω( , )=

‚xi ‚xj ‚xi ‚xj ‚xi ‚xj

√

‚ ‚ ‚ ‚

= g( , ) ’ ’1g( , )=

‚xi ‚xj ‚yi ‚xj

√ √

‚ ‚‚ ‚ ‚ ‚ ‚ ‚

= g( + , + ) ’ ’1g( ’1( ’ ), + )=

‚zi ‚ zi ‚zj ‚ zj

¯ ¯ ‚zi ‚ zi ‚zj ‚ zj

¯ ¯

98 CHAPTER 3. HODGE THEORY

aij aji aij aji

= + + ’ = aij ,

2 2 2 2

which is perfectly consistent since the matrix aij corresponds to the

hermitian structure H.

On a complex manifold M the exterior di¬eretial d can be decom-

posed as d = d + d in such a way that if ± is a di¬erential form of

pure type (p, q) then d ± and d ± are of types (p + 1, q) and (p, q + 1)

respectively. One can check that d and d satisfy the following:

d d = d d = d d + d d = 0.

Now we intend to give a method of constructing a closed 2-form ω on M

of type (1, 1) out of a smooth function f : M ’ R. A good candidate

is the form

√ √

’ ’1 n ‚ 2 f

’ ’1

ω= dd f = dzi §d¯i .

z

2 2 i=1 ‚zi ‚ zi

¯

and in many cases like in the next example we will will even get a

positive de¬nite hermitian form out of ω.

Let us take the complex projective space CPn with homogeneous

coordinates [z0 : ... : zn ] and the function f = log(|z0 |2 + · · · + |zn |2 ).

This function is not well-de¬ned on CPn as it involves homogeneous

coordinates. So we should think of f as a smooth function on Cn+1 \{0},

which is a principal bundle over CPn . Over some open contractible

open set U of CPn we can make a holomorphic choice of homogeneous

coordinates, which amounts to a section U ’ Cn+1 \ {0} of the bundle

over U . Now we have the freedom to make a holomorphic rescaling

[z0 : ... : zn ] ’ [»z0 : ... : »zn ] of the homegenous coordinates. Here

» : U ’ C— is a holomorphic function. Then the function f will change

√

to f + log |»|2 . However the form ’1d d f will not change at all since

¯

d d log |»|2 = d d (log(») + log(»)) = 0.

For example, in the case of Cn we take

n

2

|zi |2

f = |z| =

i=1

¨

3.2. KAHLER MANIFOLDS 99

to get √

’ ’1

ω= dzi §d¯i =

z dyi §dxi .

2 i i

In general, if a metric g is given by g = aij dzi d¯j , then one has

z

i,j

√ √ aij

‚‚ ‚‚ ‚‚

ω( , ) = ’g(J , ) = ’ ’1g( , ) = ’ ’1 .

‚zi ‚ zj

¯ ‚zi ‚ zj

¯ ‚zi ‚ zj

¯ 2

The function f which a symplectic form ω is obtained from is called

a K¨hler potential. Certainly, a K¨hler potential is not unique. An

a a

addition of any function f1 satisfying d d f1 = 0 will not change the

form ω.

If we return to the example of CP1 covered by open sets U0 and U1

with holomorphic coordinates z and w respectively, then we are able to

see that the symplectic form

√ √

’ ’1 dz§d¯ z ’ ’1 dw§dw ¯

ω= =

2 (1 + |z|2 )2 2 (1 + |w|2 )2

comes from a K¨hler potential

a

‚2f 1

2

f = log(1 + |z| ), since = .

¯ (1 + |z|2 )2

‚z‚ z

If we have a compact symplectic manifold (M, ω) of dimension 2n

then the 2n-form n

ω§ω§ · · · §ω

n!

is a nowhere vanishing volume form on M and hence

n

ω§ω§ · · · §ω

= 0.

n!

M

It follows that the form ω cannot be exact (otherwise the form ω§ω§ · · · §ω

n

would be exact too and by Stokes™ theorem it contradicts the above

inequality). Thus, the form ω de¬nes a non-zero cohomology class

100 CHAPTER 3. HODGE THEORY

[ω] ∈ H 2 (M, C). As a consequence we notice that if (M, ω) is a com-

pact symplectic manifold, then dim H 2 (M, C) > 0. Besides, we have

deduced that for a compact K¨hler manifold there is no global K¨hler

a a

potential f , because then d d f would be an exact form.

Returning back to CP1 we see that two potentials f0 = log(1 + |z|2 )

and f2 = log(1 + |w|2 ) de¬ned on open sets U0 and U1 correspondingly

have the property that their di¬erence f0 ’ f1 de¬ned on U0 © U1 is in

the kernel of d d (i.e., it is a harmonic function). It is so, because

f0 ’ f1 = log |z|2

is a harmonic function.

For CPn it is possible to give the same type of construction. As usual

we represent CPn = ∪n Um , where Um is the open subset on which

m=0

the m-th homogeneous coordinate does not vanish. We have natural

tk

identi¬cations Um Cn via [t0 : · · · : tn ] ’ (z1 , ..., zn ), where zk = .

tm

This also gives us coordinates (z1 , ..., zn ) on Um . On Um we have a

K¨hler potential

a

|zj |2 ).

fm (z1 , ..., zn ) = log(1 +

j

Let us explicitly notice that for this potential we have

‚ 2 fm δij zi zj

¯

= ’ .

1 + l |zl |2 (1 + l |zl |2 )2

‚zi ‚zj

So when i = j

|zl |2

‚ 2 fm 1+ l=i

= .

‚zi2 |zl |2 )2

(1 + l

‚ 2 fm

Let us take n = 2, when the matrix is given by

‚zi ‚zj

1 + |z2 |2

1 ’¯1 z2

z

.

1 + |z1 |2

’z1 z2

¯

(1 + |z1 |2 + |z2 |2 )2

This matrix is clearly positive de¬nite since its determinant is equal to

1 + |z1 |2 + |z2 |2 .

¨

3.2. KAHLER MANIFOLDS 101

We produce the following chain of arguments to show that this

matrix is positive de¬nite over Um for general n. The associated two-

√

’ ’1

forms ωm = 2 d d fm and ωp agree over the intersection Um © Up ,

therefore they glue into a global 2-form ω. The unitary group U (n+1) ‚

GL(n + 1, C) acts on CPn preserving ω, because U (n + 1) preserves the

norm i |zi |2 on Cn+1 . The action of the group U (n + 1) is transitive on

CPn , hence it is enough to show that the candidate for our Riemannian

metric g(v, w) = ω(Jv, w) is positive de¬nite at one point. Since ω and

J are U (n+1)-invariant, the metric g itself is invariant too. Let us take

Cn . If we evaluate the

the origin of U0 under the identi¬cation U0

2-form ω at zero we will see that the corresponding hermitian metric is

just the identity matrix, which is positive de¬nite.

Now we are going to give direct geometric construction of a her-

mitian metric H on CPn . We start with the norm ||z||2 = i zi zi ¯

on Cn+1 , which clearly de¬nes a hermitian inner product on Cn+1 by

(z, w) = i zi wi . The tangent space to CPn at some point corresponding

¯

to a line l is Tl CPn = Hom(l, l⊥ ), where l⊥ is the orthogonal comple-

ment to l: Cn+1 = l • l⊥ . This description of the tangent space can be

explained as follows. We represent a tangent vector to CPn by smooth