<<

. 16
( 44 .)



>>

‚zi 2 ‚xi ‚yi ‚ zi
¯ 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

<<

. 16
( 44 .)



>>