· · · ’ Aj’1 (M ) ’ Aj (M ) ’ Aj+1 (M ) ’ · · · .

Similarly, the complex-valued cohomology groups H j (M, C) = H j (M, R)—R

C are computed by the complex

d d d d

· · · ’ Aj’1 (M ) — C ’ Aj (M ) — C ’ Aj+1 (M ) — C ’ · · · .

If ± is a j-form then we can decompose it with respect to types ± =

p,q p,q

p+q=j ± , where ± is a form of type (p, q). One of the main results

for the cohomology of K¨hler manifolds is the Hodge decomposition

a

which says that

H j (M, C) = H p,q ,

p+q=j

where the space H p,q consists of classes in H j (M, C) represented by

a closed j-form of type (p, q). Moreover the decomposition does not

depend upon a choice of the K¨hler form.

a

The ¬rst and simplest situation which we want to consider in detail

is the case when M is a compact Riemann surface (compact complex

manifold of dimension 1) of genus g (number of holes in M ). Let us

choose a representation of a basis in H1 (M, Z) by cycles (a1 , ..., ag , b1 , ..., bg )

on M in such a way that the matrix Π of intersection pairing will be

0 ’I

Π= , ai , bj = δij , ai , aj = bi , bj = 0.

I0

Let „¦1 (M ) stand for the space of global holomorphic 1-forms on M .

Locally any ω ∈ „¦1 (M ) can be expressed as ω = f (z)dz for a holo-

morphic function f (z). We note that in general, if ± = F dz, where

F is a C ∞ -function on a Riemann surface M , then ± is closed if and

only if F is a holomorphic function. It is easy to see that moreover ±

is exact only if F = 0. Indeed if ± = dg, then g must be a holomorphic

function; but since M is compact any global holomorphic function is

108 CHAPTER 3. HODGE THEORY

constant. Thus we conclude that „¦1 (M ) = H 1,0 (M ) and the dimension

of this space of global holomorphic 1-forms on M is g.

For instance, let M ‚ CP2 be given in homogeneous coordinates

[X0 : X1 : X2 ] as the zero locus of a homogeneous polinomial F (X0 , X1 , X2 )

of degree d. If the polynomials ‚F/‚X0 , ‚F/‚X1 , ‚F/‚X2 have no

common zero except (0, 0, 0) then M is smooth and M is a compact

Riemann surface of genus (d ’ 1)(d ’ 2)/2. We will try to give an ex-

plicit basis of „¦1 (M ). Let us take the open set U2 ‚ CP2 , the set U2 is

given by X2 = 0, and let h(w0 , w1 ) = F (w0 , w1 , 1). Then „¦1 (M ) is the

space of holomorphic di¬erentials

P (w0 , w1 )dw0

ω= ,

‚h/‚w1

where P is a polynomial of degree ¤ d ’ 3. As an example let us

2 3

consider the Weierstraß cubic curve F = X2 X1 ’ X2 Q(X0 /X2 ), where

Q is a polynomial of degree 3. It corresponds to the case g = 1, here

2

h(w0 , w1 ) = w1 ’ Q(w0 ), and hence

dw0

ω= .

2w1

The dimension of the space H 1 (M, C) over C is 2g and this space is

equipped with the symplectic form

S([±], [β]) = ±§β ∈ C.

M

It is true that H 1,0 (M ) = „¦1 (M ) ‚ H 1 (M, C) is a Lagrangian subspace.

(As usual a subspace A of a symplectic vector space B with a symplectic

form ω is called isotropic if for any a1 , a2 ∈ A one has ω(a1 , a2 ) = 0. The

subspace A is called Lagrangian if it is a maximal isotropic subspace, or,

equivalently, A is an isotropic subspace and dim(A) = dim(B)/2. This

is equivalent to A = A⊥ with respect to ω.) The space H 0,1 ‚ H 1 (M, C)

is complex-conjugate to „¦1 (M ) and consists of the anti-holomorphic 1-

forms, i.e. the 1-forms ± that locally can be written as ± = g(z)d¯, z

where g(z) is an anti-holomorphic function (dg/dz = 0).

LEMMA 3.4.1

H 1 (M, C) = „¦1 (M ) • „¦1 (M ).

3.5. HARMONIC FORMS ON COMPACT MANIFOLDS 109

¯

Proof. Let β ∈ „¦1 (M ), β = 0. Then locally β = f (z)dz and β =

√

¯z ¯

f (z)d¯, hence β§β = |f (z)|2 dz§d¯. We notice that dz§d¯ = ’2 ’1dx§dy,

z z

√

where z = x + ’1y. Since dx§dy is a volume form on M and f is

not identically zero, we get

√

¯

’1 β§β > 0.

¯

Suppose now that [β] ∈ H 0,1 as well. This would imply that [β] ∈ H 1,0 .

¯

Since H 1,0 is a Lagrangian subspace, this would mean that β§β = 0,

which contradicts our previous observation. Now, it remains to be

noticed that

2g = dim H 1 (M, C) = dim „¦1 (M ) + dim „¦1 (M ) = g + g.

3.5 Theory of harmonic forms on compact

riemannian manifolds

The basic idea leading to the Hodge theorem is to represent cohomology

classes by ”harmonic forms”. We will not provide the analytical details

that are contained in the proof this result1 , instead we will underscore

the major points. When M is a compact riemannian manifold, we are

given a positive de¬nite symmetric bilinear form g on T M as well as

on its dual T — M and its exterior powers §j T — M . Thus on Aj (M ) we

have a positive de¬nite inner product

±, β = g(±x , βx )ν,

M

where ν is the volume form. In local coordinates (x1 , ..., xn ) on M we

have ν = det(gij )dx1 § · · · §dxn , where (gij ) is the metric tensor.

It is a known fact that there exists a di¬erential operator δ of order

1, which is a formal adjoint to d

δ : Aj (M ) ’ Aj’1 (M ),

1

Those who are interested can consult either the book of Ph. Gri¬ths and J.

Harris, Principles of Algebraic Geometry [31], or a book of F. Warner, Foundations

of Di¬erentiable Manifolds and Lie Groups [55].

110 CHAPTER 3. HODGE THEORY

d±, β = ±, δβ , ± ∈ Aj’1 (M ), β ∈ Aj (M ).

Let us consider a special case when M is a torus M = Tn = Rn /Zn

with standard riemannian metric. Let (x1 , ..., xn ) be the coordinates

on Rn , then

j

‚f

(’1)l

δ(f dx1 § · · · §dxj ) = ’ dx1 § · · · §dxl § · · · §dxj ,

‚xl

l=1

where the hat is placed over the omitted di¬erential. It is easy to check

that this is formally adjoint to d using the following consequence of

Gauß™ theorem:

‚f ‚g

gν=’ f ν.

M ‚xi M ‚xi

Another basic fact is that the degree j cohomology of M is isomor-

phic to Ker(d : Aj ’ Aj+1 ) © Ker(δ : Aj ’ Aj’1 ).

LEMMA 3.5.1 Ker(d) © Ker(δ) = Ker(∆), where ∆ is the Laplace

operator or Laplacian de¬ned by ∆ = dδ + δd.

Proof. We only need to show that ∆± = 0 implies both d± = 0 and

δ± = 0. We have

0 = ±, ∆± = ±, dδ± + ±, δd± = δ±, δ± + d±, d± .

Since the pairing , is positive de¬nite we have that both δ±, δ± = 0

and d±, d± = 0, which implies the desired result.

In the case of Tn we have

∆(f dx1 § · · · §dxj ) = (∆f )dx1 § · · · §dxj ,

where n

‚ 2f

∆f = ’ .

‚x2i

i=1

When n = 1 and we have a circle T = R/Z then the Hilbert space L2 (T)

√

has orthonormal basis {exp(2π ’1rx)}, r ∈ Z, and from

√

√

2

‚ exp(2π ’1rx) 22

= ’4π r exp(2π ’1rx)

‚x2

we see that all eigenvalues of ∆ are positive real numbers.

3.5. HARMONIC FORMS ON COMPACT MANIFOLDS 111

DEFINITION 3.5.2 A j-form ± is called harmonic if ∆± = 0, or,

equivalently, d± = δ± = 0.

THEOREM 3.5.3 (Hodge.) If M is a compact oriented rieman-

nian manifold then the space of real harmonic j-forms is isomorphic to

H j (M, R).

Alternatively, the space of complex harmonic j-forms is isomorphic

to H j (M, C). The space of all j-forms has the following structure

Aj (M ) = I • II • III,

where I is the space of harmonic j-forms, II = Im(d), and III =

Im(δ). The forms in the space III are called coexact forms. The

direct sum I • II forms the space of closed forms, I • III is the space

of coclosed forms and II • III corresponds to the positive spectrum of

∆.

On a compact riemannian manifold M (with riemannian metric g)

of dimension n for the operator δ : Aj (M ) ’ Aj’1 (M ) one has

g(±, δβ)ν, d±, β ∈ Aj (M ).

g(d±, β)ν =

M M

The operator δ can be de¬ned even for non-compact manifolds, but in

this case the formula above is true only when ± and β are compactly

supported forms.

The next step is to consider the Hodge star operator

— : Aj (M ) ’ An’j (M ), ±§(—β) = g(±, β)ν, ±, β ∈ Aj (M ).

The operator — is of order zero and it is purely “punctual”, meaning

that one can de¬ne it pointwise:

—x : §j Tx M ’ §n’j Tx M.

— —

We de¬ne it on a basis of Aj (M ). Let J = {i1 < ... < ij } be a subset of

(1, ..., n), and let K = {k1 < ... < kn’j } be the complementary subset.

Let µ be the sign of the permutation (i1 , ..., ij , k1 , ..., kn’j ). If e1 , ..., en

—

is an oriented orthonormal frame for Tx M , then by de¬nition

—ei1 § · · · §eik = µek1 § · · · §ekn’j .

112 CHAPTER 3. HODGE THEORY

We note that —2 as a linear automorphism of Aj (M ) is equal to (’1)pn+p Id.

The sign (’1)pn+p is that of the permutation (j +1, · · · , n, 1, · · · , j). Be-

sides, it is easy to check that — is a unitary operator.

In the case n = 2k we have — : Ak (M ) ’ Ak (M ), —2 = (’1)k Id.

When k is even Ak (M ) = SD • ASD is represented as a sum of two

spaces of equal dimension consisting of self-dual and anti-self-dual forms

which are the eigenspaces corresponding to the eigenvalues 1 and ’1 of

√

—2√respectively. When k is odd then the eigenvalues of —2 are ’1 and

’ ’1, so one has a complex structure on the vector bundle §k Tx M . —

LEMMA 3.5.4 On the space Aj (M ) we have

δ = (’1)n+pn+1 — d — .

Proof. Let ± ∈ Aj’1 (M ) and β ∈ Aj (M ). Then integrating by parts

d±§(—β) = (’1)p

d±, β = g(d±, β)ν = ±§d(—β) =

M M M

= (’1)p g(±, —’1 d — β)ν = (’1)p (’1)(p’1)(n’p’1) g(±, —d — β)ν.

M M

In particular, when n is even one gets δ = ’ — d—. We also notice

that —1 = ν - the volume form.

3.6 Hodge theory on K¨hler manifolds

a

Let us ¬rst make detailed computations in complex dimension 1 (n =

√

2). Locally we have complex coordinate z = x + ’1y and a metric

g = f (x, y)(dx2 + dy 2 ), f (x, y) > 0, and the volume form ν = f dx§dy.

We will show how δ acts on 1-forms. We have

—(hdx) = hdy, —(mdy) = ’mdx, h, m ∈ C ∞ (M ).

Since —ν = 1 and —(dx§dy) = 1/f , one has

‚h 1 ‚h

δ(hdx) = ’ — d — (hdx) = ’ — d(hdy) = ’ — ( dx§dy) = ’

‚x f ‚x