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w(z) = u + ’1v. Here the Cauchy-Riemann condition amounts to
‚u ‚v ‚u ‚v
= ‚y and = ’ ‚x . The corresponding di¬erential of the mapping
‚x ‚y

‚u ‚v
a ’b
‚x ‚y
dz w = = ’ C ‚ M (2, R).
‚u ‚v
ba
‚y ‚x

Note that C is viewed as a subalgebra of M (2, R) by making √ com-
a
plex number acting by multiplication on C = R2 , with basis (1, ’1).
Returning to the general situation, the existence of an almost com-
plex structure M does not guarantee that there exists locally an embed-
ding U ’ Cn , where U ‚ M , compatible with the complex structure.
The obstruction to ¬nd such an embedding is the Nijenhuis tensor S
of type (2, 1). For two real vector ¬elds ξ and · it is

S(ξ, ·) = [ξ, ·] ’ [J(ξ), J(·)] + J[J(ξ), ·] + J[ξ, J(·)].

In ¬nite dimensions, a famous theorem of Newlander and Nirenberg
[46] asserts that an almost complex structure J is integrable if and
only if the Nijenhuis tensor vanishes identically. In real dimension 2
every almost complex structure is integrable, since S(ξ, Jξ) = 0.
If we have a real oriented manifold M of dimension 2, it is called a
Riemann surface. A choice of riemannian structure de¬nes a complex
structure as we shall see in a moment. For any x ∈ M we have Tx M
R2 ; now for any v ∈ Tx M , the vector Jv is de¬ned uniquely by the
following set of conditions. First we require that v is perpendicular to
Jv, secondly, they have the same norm: ||v|| = ||Jv||, and ¬nally, we
need the pair (v, Jv) to form an oriented basis of Tx M .
It is very natural to ask how many riemannian structures give rise
to the same complex structure. If we rescale a given riemannian metric
by a smooth function » : M ’ R— taking positive real values, then the
+
induced complex structure will certainly be the same. Two metrics g
(which is a positive de¬nite symmetric bilinear map T X — T X ’ R)
and »g give rise to the same J, they are called conformally equivalent.
Thus a conformal structure on a manifold is a metric up to rescaling.
As we saw, on an oriented surface a complex structure compatible with
the orientation is the same as conformal structure. If we have a map
f : M ’ N between two oriented surfaces endowed with metrics gM
and gN respectively, then it is holomorphic if and only if it is a conformal
92 CHAPTER 3. HODGE THEORY

map, meaning that two metrics on M : gM and f — gN are conformally
equivalent, and f preserves the orientation.
On C = R2 the standard riemannian structure compatible with

J = ’1 is given by g = dx2 + dy 2 . In terms of the holomorphic
coordinate z it is equal to dzd¯. The pair (dz, d¯) forms a basis of com-
z z

plexi¬ed cotangent bundle T M — C. More general riemannian metrics
compatible with J can be written as g = »(z)dzd¯, where the smooth
z
function » takes only positive values.
An important example we would like to consider involves the upper-
half plane H = {z ∈ C; Im(z) > 0} (also called Lobachevsky plane)
with the hyperbolic metric
dzd¯
z ’4dzd¯ z
g= = .
Im(z)2 (z ’ z )2
¯
The group P SL(2, R) = SL(2, R)/(±1) acts on H by fractional-linear
transformations
az + b ab
γz = ; γ= .
cd
cz + d
PROPOSITION 3.1.1 The group P SL(2, R) acts on H by isome-
tries.
Proof. We notice that
Im(z) d(γz) 1
Im(γz) = , and = .
|cz + d|2 (cz + d)2
dz
Let us transform g under γ ∈ SL(2, R):
|cz + d|4
d(γz)d(γz) dz d¯
z dzd¯
z
gγ = = = .
Im(γz)2 (cz + d)2 (c¯ + d)2 Im(z)2 Im(z)2
z

Another essential example appears if one takes S = C ∪ {∞} - the
Riemann sphere. We represent S = U0 ∪ U1 as the union of two open
subsets each of which is di¬eomorphic to C. On U0 and U1 we have
coordinates z and w respectively satisfying the identity zw = 1 over
the intersection U0 © U1 . The point at in¬nity for the coordinate z
corresponds to the value w = 0. Let us take the metric
dzd¯z
g= ,
(1 + |z|2 )2
¨
3.2. KAHLER MANIFOLDS 93

which is invariant under the substitution w = 1/z, as is easily seen
using the identities
1 + |z|2
dw dwdw ¯
= |z|2 .
dz = ’ 2 , dzd¯ =
z , and
4 2
w |w| 1 + |w|
Thus we have a global riemannian metric g on S. The group of isome-
tries for this metric is the Lie group P SU (2) = SU (2)/(±1). It may
be viewed as a subgroup of the group of biholomorphic maps S ’ S
which is SL(2, C)/(±1) given by
a ’¯ b
; a, b ∈ C, |a|2 + |b|2 = 1}/(±1).
{
ba ¯
(We notice also that SU (2) is maximal compact subgroup of SL(2, C).)
Another way to look at the compatibility condition is to say that J
is an orthogonal transformation with respect to g: g(Jv, Jw) = g(v, w).
Indeed, it automatically follows that |Jv| = |v| and
g(v, Jv) = g(Jv, J —¦ Jv) = g(Jv, ’v) = ’g(v, Jv) = 0.
We will call such a Riemannian metric compatible with a complex struc-
ture hermitian.
LEMMA 3.1.2 For any complex manifold M there exists a hermitian
metric g.
The proof of this Lemma is a standard application of a partition of
unity. First, it is possible to construct a hermitian metric on each open
set of some cover {Ui } of M and then do it globally using a partition
of unity subordinate to this open cover {Ui }.
It is a nice exercise to describe the set of all hermitian metrics
compatible with a given complex structure. And, conversely, given any
riemannian metric g, describe the set of compatible complex structures.


3.2 K¨hler manifolds
a
We recall a hermitian form on a complex vector space V . It is a
sesquilinear form H(v, w) additive in v and w , taking values in C
and satisfying the properties
H(»v, µw) = »¯H(v, w);
µ
94 CHAPTER 3. HODGE THEORY

H(v, w) = H(w, v),
or, equivalently,
H(v, v) ∈ R.
A hermitian form H is called positive de¬nite if for any v = 0 one has
H(v, v) > 0. When V = Cn any hermitian form H can be represented
¯
by a n — n matrix A = (aij ) with the property A = At . In other words,
the matrix A is hermitian. If we represent two vectors v, w ∈ Cn by
columns «  « 
v1 w1
¬v · ¬w ·
¬ 2· ¬ ·
v = ¬ ·, w = ¬ 2 ·,
··· ···
vn wn
then we have
H(v, w) = aij vi wj .
¯
1¤i,j¤n

Sometimes it is convenient to use Sylvester™s theorem which asserts
that an n — n hermitian matrix A = (aij ) is positive de¬nite if and only
if n diagonal minors are positive reals:

a11 > 0, a11 a22 ’ a21 a12 > 0, ..., det(A) > 0.

Now the operator J corresponds to the multiplication by ’1 on
V and let us decompose

H(v, w) = g(v, w) + ’1ω(v, w).

In view of the fact that
√ √
H(w, v) = H(v, w) = g(v, w) ’ ’1ω(v, w) = g(w, v) + ’1ω(w, v)

we obtain that the real part of the hermitian form g(v, w) = g(w, v)
is symmetric bilinear and the imaginary part ω(v, w) = ’ω(w, v) is
skew-symmetric. The relation between g and ω can be obtained from
the following consideration:

g(Jv, w) + ’1ω(Jv, w) = H(Jv, w) =
√ √
= ’1H(v, w) = ’ω(v, w) + ’1g(v, w),
¨
3.2. KAHLER MANIFOLDS 95

thus we get g(v, w) = ω(Jv, w). Besides, from the equality H(Jv, Jw) =
H(v, w) re¬‚ecting the fact that J is orthogonal for H we see that J is
orthogonal for g and ω as well:

g(Jv, Jw) = g(v, w); ω(Jv, Jw) = ω(v, w).

So, we have established

PROPOSITION 3.2.1 Let V be a vector space over C. Then there
is a linear bijection between
(1) the real vector space of Hermitian forms H on V
(2) the real vector space of symmetric bilinear forms g on V such that

g(Jv, Jw) = g(v, w).

Furthemore, H is positive de¬nite if and only if g is.

Now we shall de¬ne a symplectic structure on a manifold M . It
means that on M there exists a non-degenerate closed 2-form ω. The
form ω is called the symplectic form. The condition that form ω is
closed simply means that its exterior derivative vanishes: dω = 0. By
de¬nition, ω provides us with a skew-symmetric pairing Tx M —Tx M ’
R and the non-degeneracy condition is that ω(v, w) = 0 for all w if and
only if v = 0. The basic result in the theory of symplectic manifolds is
the Darboux theorem which tells us that on a symplectic manifold M of
dimension 2n (each symplectic manifold is even-dimensional) there are
locally coordinates (p1 , ..., pn , q1 , ..., qn ) called Darboux coordinates such
that the symplectic form is ω = n dpi §dqi in those local coordinates.
1
The ¬rst example to consider is the ”¬‚at” example of the manifold

Cn with coordinates zi = xi + ’1yi . For each z ∈ Cn the tangent space
is Tz Cn = Cn and let us take the hermitian form H(v, w) = n vi wi . It ¯
1
is easy to see that in this case g = Re(H) = n dx2 + n dyi is the 2
1 i 1
standard metric on R2n and ω = n dyi §dxi is the standard so-called
1
Liouville symplectic form. Passing to the coordinates zi and taking into
account that
dzi + d¯i
z dzi ’ d¯i
z

dxi = ; and dyi =
2 2 ’1
96 CHAPTER 3. HODGE THEORY

’1 n
we can rewrite the Liouville symplectic form as ω = 2 1 dzi §d¯i .
z
We say in this situation that ω is of pure type (1, 1). In general we
say that a di¬erential form ± is of type (p, q) if it can be written down
as ± = I,J fIJ dzI d¯J , where I and J as multiindices are subsets of
z
{1, 2, ..., n} of ¬xed cardinality #I = p and #J = q.
Now we are ready to give an important de¬nition.
DEFINITION 3.2.2 A K¨hler manifold is a complex manifold (M, J, H)
a
endowed with a hermitian form H such that its real part g = Re(H) is
a riemannian metric on M and its imaginary part ω = Im(H) de¬nes
a symplectic structure on M .
In fact one can notice that a hermitian form H is non-degenerate if and
only if Im(H) is such. Another general fact about hermitian manifolds
is that the form ω = Im(H) is always of type (1, 1). It is called the
K¨hler form. Moreover, we can establish
a
LEMMA 3.2.3 If (M, J) is a complex manifold endowed with a real
two-form ω then the following two conditions are equivalent
(1) There is a hermitian form H on M such that Im(H) = ω.
(2) The form ω is of type (1, 1).
Proof. In local coordinates (z1 , ..., zn ) on a complex manifold (M, J)
we have
√ √
‚ 1‚ ‚ ‚ 1‚ ‚
=( ’ ’1 ), and =( + ’1 ).

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