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 ).