dimC H p (M, C) = 2 i<j dim H i,j (M ).

For the second proof we construct a non-degenerate bilinear form

Q on H p (M, R). It is enough to do this for p ¤ n since

dim H p (M, R) = dim H 2n’p (M, R).

Let ±, β be two closed di¬erential forms representing classes [±], [β]

∈ H p (M, R). We set

±§β§ω n’p .

Q([±], [β]) = ±

M

This form is skew-symmetric when p is odd and symmetric when p

is even. Assume for a moment that [β] is perpendicular to [±] for

any p-form ±. It follows that M ±§(β§ω n’p ) = 0 for any closed ±.

This can only happen if β § ω n’p represents zero cohomology class in

H 2n’p (M, R). In view of Hard Lefschetz [β] = 0 as well.

As a simple example, let us consider the case of M = C2 /Λ, where

Λ is a complete lattice. We will show how the Hard Lefschetz Theorem

L

works when applied to H 1 (M, C) ’ H 3 (M, C), where L = c(dz1 §d¯1 +

z

√

’1

dz2 §d¯2 ), c = ’ 2 . The basis of H 1 (M, C) is constituted by

z

dz1 , dz2 , d¯1 , d¯2 .

z z

It is mapped by L to a basis

cdz1 §dz2 §d¯2 , cdz1 §dz2 §d¯1 , cd¯1 §dz2 §d¯2 , cdz1 §d¯2 §d¯1 .

z z z z z z

Let V be a representation of sl(2, C).

DEFINITION 3.6.21 The primitive cohomology P rim’k (V ) ‚ V ’k

consists of those vectors v of veight ’k which satisfy Y v = 0, or, equiv-

alently, X k+1 v = 0.

PROPOSITION 3.6.22 For any V we have

V ’k = P rim’k (V ) • X · P rim’k’2 (V ) • X 2 · P rim’k’4 (V ) • · · · .

126 CHAPTER 3. HODGE THEORY

Proof. Straightforward.

Now we apply this Proposition to the cohomology of a compact

K¨hler manifold M . We de¬ne

a

Hprim (M )=Ker[Λ : H k (M, C) ’ H k’2 (M, C)] =

k

= Ker[Ln’k+1 : H k (M, C) ’ H 2n’k+2 (M, C)].

We get

H k (M, C) = Hprim (M ) • L · Hprim (M ) • L2 · Hprim • · · · .

k k’2 k’4

This allows us to conclude that

dim H n (M, C) ≥ dim H n’2 (M, C) ≥ dim H n’4 (M, C) ≥ · · ·

and

dim H n’1 (M, C) ≥ dim H n’3 (M, C) ≥ dim H n’5 (M, C) ≥ · · · .

EXAMPLE of a non-closed holomorphic 1-form ω on a compact com-

plex manifold M . This example is also important because it is simple

and it immediately provides us with a compact complex manifold M

which is not K¨hler because we proved before that on K¨hler manifolds

a a

every global holomorphic k-form is in fact closed (and never exact un-

less it is the zero form). Let H be the complex Heisenberg group of

3 — 3 matrices of the following form

«

1xz

¬ ·

0 1 y , x, y, z ∈ C.

001

We also think of x, y, z as complex coordinates de¬ning the complex

manifold structure on this complex a¬ne non-commutative Lie group.

The group multiplication is just matrix multiplication and is given by

« « «

1xz 1ac 1 a + x c + xb + z

¬ ·¬ · ¬

y + b ·.

0 1 y · 0 1 b = 0 1

001 001 0 0 1

¨

3.6. HODGE THEORY ON KAHLER MANIFOLDS 127

If such two elements of H commute, then xb = ya and hence the center

of H is isomorphic to C and consists of those matrices which have

x = y = 0.

LEMMA 3.6.23 The form ω = dz ’ ydx is a right-invariant non-

closed holomorphic 1-form on H.

To proof this assertion we simply check that the e¬ect of translation of

ω by an element «

1ac

¬ ·

0 1 b ∈ H

001

transforms it into

d(z + c + bx) ’ (b + y)d(a + x) = dz + bdx ’ bdx ’ ydx = ω.

Also we see that

dω = ’dy§dx = dx§dy = 0.

Now we de¬ne M as the quotient H/“, where “ is the following

discrete cocompact subgroup in H.

«

1ac √

¬ ·

“ = { 0 1 b , a, b, c ∈ Z[ ’1]},

001

√

where Z[ ’1] is the ring of so-called Gaussian integers, i.e. the complex

√

numbers of the form n + ’1m, m, n ∈ Z. To explain the compactnes

of H/“ we just point out the following nice ¬bration

«

1xz

√ ¬ ·

C/Z[ ’1] ’ H/“ 0 1 y

001

“ “

√

(C/Z[ ’1])2 (x, y)

Therefore, due to the fact that ω is right-invariant, it descends to a

non-closed holomorphic 1 form on H/“ - a compact complex manifold,

which is therefore not K¨hler .

a

128 CHAPTER 3. HODGE THEORY

3.7 Hodge Conjecture

A Hodge structure of weight j (over R) consists of a real vector space

V together with a decomposition of its complexi¬cation

V — C = •p+q=j V p,q

such that V p,q = V q,p .

Obviously, the group H j (M, R) for a compact K¨hler manifold M

a

has a Hodge structure of weight j. In this situation we would like to

give an intrinsic description of groups H p,q ‚ H p+q (M, C), i.e. such

a description which does not use metric or K¨hler form, but only the

a

complex structure on M .

LEMMA 3.7.1 The group H p,q (M ) consists of those classes of com-

plex coe¬cient closed p + q-forms on M that have representatives of

type (p, q).

Proof. What really needs to be shown is that if ± is a closed form

on M of pure type (p, q) then there exists a harmonic form β on M

of type (p, q) such that ± ’ β is an exact form. In this case we call

β the harmonic representative of ±. The space Hp+q (M ) of harmonic

forms of total degree p + q is a ¬nite-dimensional Hilbert subspace of

the pre-Hilbert space Ap+q (M ) of closed (p + q)-forms on M . We note

that the orthogonal space to Hj in the space of closed j-forms is the

space of exact j-forms. Indeed if ± is harmonic and β is exact, say

β = dγ, then

±, β = ±, dγ = δ±, γ = 0.

Since the dimension of Hj (M ) is equal to the codimension of the exact

j-forms in the closed j-forms, the statement follows. Thus we can

de¬ne the orthogonal projection Q : Ap+q (M ) ’ Hp+q (M ). Now we

let β = Q(±). Then if ± is closed, ± and β are cohomologous. Since we

have an orthogonal decomposition

Aj (M ) = •p+q=j Ap,q (M )

“ •Qp,q

Q“

Hj (M ) = •p+q=j Hp+q (M )

3.7. HODGE CONJECTURE 129

It follows that if ± is of type (p, q) then β is of type (p, q) as well.

It is very important to underline the algebraic structure on coho-

mology of M .

LEMMA 3.7.2 The wedge product of di¬erential forms induces cup-

product on cohomology

∪

H j (M, C) — H k (M, C) ’ H j+k (M, C),

∪

H p,q (M ) — H r,s (M ) ’ H p+r,q+s (M ).

In this way we obtain a bi-graded algebra.

Proof. The statement follows from a simple observation that if ± is a

closed form of type (p, q) and β is a closed form of type (r, s) then ±§β

is a closed form of type (p + r, q + s).

Whenever we encounter some interesting phenomena concerning al-

gebraic structures on cohomology, we wish we were able to give a nice

geometric interpretation to them. In certain situations it is really possi-

ble to do so. For instance one can interpret the fact that when M = CPn

(n > 1) the group H 2p (CPn , C) coincides with its subgroup H p,p just by

simple dimension calculation, or, we can use our previous knowledge

that the form ω§ · · · §ω is harmonic and of type (p, p). In general,

p

classes of type (p, p) are very important, because they are intimately

related with (complex) analytic cycles (which are complex analytic sub-

varieties). Let X ‚ M be a complex analytic subvariety (possibly sin-

gular) of codimension d (M as before is a compact K¨hler manifold of

a

dimension n). There is a well-de¬ned cohomology class

[X] ∈ H 2d (M, Z) ’ H 2d (M, C),

which is de¬ned using the Poincar´ duality H i (M, Z)

e H2n’i (M, Z).

The fact that this class is still de¬ned when X is singular is a remarkable

theorem of Bloom-Herrera.

PROPOSITION 3.7.3

[X] ∈ H d,d (M )

130 CHAPTER 3. HODGE THEORY

Proof. The Poincar´ duality pairing between H 2d (M, C) and H 2n’2d (M, C)

e

respects the Hodge decompositions

H 2d (M, C) = •p H p,2d’p

and

H 2n’2d (M, C) = •q H q,2n’2d’q .

We claim that H p,2d’p is perpendicular to H q,2n’2d’q unless p + q = n.

Indeed, if ±, β are of types (p, 2d ’ p) and (q, 2n ’ 2d ’ q) respectively,

then ±§β is of type (p + q, 2n ’ p ’ q) and falls out of the picture

if p + q = n. Thus we coclude that H p,2d’p pairs non-singularly with

H n’p,n’2d+p .

Let κ be a form of type (r, 2n ’ 2d ’ r). By de¬nition of the duality

pairing we have

[X], [κ] = κ|X .

X

So, if r = n ’ d then the forms κ restricts to zero on X. Using non-

degeneracy of the pairing we arrive to the fact that [X] is of pure type

(d, d).

The statement converse to the one in Proposition is known as the

Hodge conjecture and is still open after more than 50 years of almost

fruitless attempts. We say that [Y ] ∈ H 2p (M, C) is a Hodge cohomology

class if it is integral (i.e. coming from H 2d (M, Z)) and is purely of type

(d, d).

THEOREM 3.7.4 (Lefschetz). Let X ‚ CPn be a projective mani-

fold. Then any Hodge class κ of degree 2 comes from complex-analytic

subvarieties (possibly singular) X1 , ..., Xk of codimension 1 in X, i.e.

κ = k ±[Xi ].

i=1

The proof of this result can be found e.g. in [31]. This theorem estab-

lishes the Hodge conjecture for the case d = 1, but unfortunately this is

the only case proved in general. In its ¬rst form the Hodge conjecture

was formulated as the above theorem when one changes 2 to 2d. But

after Atiyah and Hirzebruch found a counterexample in the 1960™s, the

conjecture was modi¬ed and is known at the present in the following

form.

3.8. HODGE DECOMPOSITION AND SHEAF COHOMOLOGY131

Hodge Conjecture. Let X be a projective manifold, and let κ be a

Hodge class κ of degree 2d. Then there exist complex-analytic subvari-

eties (possibly singular) X1 , ..., Xk of codimension d in X and an integer

h such that

k

h·κ= ±[Xi ].

i=1