ńņš. 22 |

3.8 Hodge decomposition and sheaf coho-

mology

Now let us give a couple of diļ¬erent interpretations of the groups

H p,q (M ) arising in the Hodge decomposition

H j (M, C) = ā•p+q=j H p,q (M )

of cohomology of a compact KĀØhler manifold M . We already know

a

that H p,q (M ) is the space of harmonic forms of type (p, q), or, equiva-

lently, it is the space of cohomology classes in H j (M, C) which have a

representattive diļ¬erential form of type (p, q). Now we give the third

description which uses Dolbeault cohomology. Consider the complex

d d d

Ap,0 (M ) ā’ Ap,1 (M ) ā’ Ā· Ā· Ā· ā’ Ap,n (M ),

where n = dimC M and Ap,q (M ) is the space of complex-valued diļ¬er-

ential forms on M of type (p, q).

PROPOSITION 3.8.1 The degree q cohomology of this Dolbeault

complex identiļ¬es with the space H p,q .

For instance, we have

PROPOSITION 3.8.2 For Ī± ā Ap,0 (M ), Ī± is holomorphic if and

only if d Ī± = 0.

COROLLARY 3.8.3

Ker[d : Ap,0 (M ) ā’ Ap,1 (M )] = ā„¦1 (M ) = H p,0 (M ).

132 CHAPTER 3. HODGE THEORY

One can, actually, analyze the Dolbeault complex using harmonic forms

with respect to the anti-holomorphic Laplace operator ā . We know

that ā = ā/2, hence the spaces of diļ¬erential forms Ī± of type (p, q)

such that ā Ī± = 0 and āĪ± = 0 coincide. We now give a brief account

on two others interpretations of the space H p,q (M )

In terms of sheaf cohomology H p,q (M ) = H q (M, ā„¦p ), where ā„¦p denotes

the sheaf of holomorphic p-forms.

Ė Ė

In terms of Cech cohomology H p,q (M ) = H q (M, ā„¦p ).

The equivalence of those descriptions follows from the theorem which

identiļ¬es sheaf cohomology with coeļ¬cients in holomorphic sections of

Ė

a vector bundle with Cech cohomology with coeļ¬cients in this bundle

for all reasonable manifolds (e.g. compact).

3.9 Formality of cohomology of compact

KĀØhler manifolds

a

In this section we deal with a compact KĀØhler manifold M unless oth-

a

erwise stated. By Ap,q (M ) we understand the space of complex-valued

diļ¬erential forms on M of type (p, q).

We recall the Hodge theorem which states that the space of har-

monic j-forms on M is isomorphic to H j (M, C). Moreover, H j (M, C)

decomposes into the direct sum of spaces Hp,q (M ) of harmonic forms

of type (p, q), where p + q = j. Let H stand for the operator of the

orthogonal projection Ap,q (M ) ā’ Hp,q (M ). Since all the cohomology

groups of M are ļ¬nite-dimenional, H is well-deļ¬ned. We also recall

that the space Ap,q (M ) is the direct sum of Hp,q (M ) and ā(Ap,q (M )).

p,q

Let Hā„ (M ) stand for the orthogonal complement of the space of har-

monic forms of type (p, q) in Ap,q (M ). It identiļ¬es then with the image

of Ap,q (M ) under the action of ā. It also follows that the operator ā

p,q

restricted to Hā„ (M ) is invertible.

Let us deļ¬ne the Greenā™s operator G : Ap,q (M ) ā’ Ap,q (M ) as a

composition

G = (ā|Hā„ (M ) )ā’1 ā—¦ (Id ā’ H),

p,q

3.9. FORMALITY OF COHOMOLOGY 133

where Id is the identity operator. The basic properties of the Greenā™s

operator G are summarized in

PROPOSITION 3.9.1 The operators G and H commute with every

linear map T : Ap,q (M ) ā’ Ap,q (M ) which commutes with ā. Besides,

one has:

Hā = āH = 0, GH = HG = 0, Id = H + āG = H + Gā.

We leave the proof of this Proposition as an easy exercise to the reader.

One of the other properties of G that we brieļ¬‚y mention is that the

operator G is continuous (in the natural FrĀ“chet topology on Ap,q (M )).

e

Analogously one can deļ¬ne the Greenā™s operators G and G associ-

ated to the Laplace operators ā and ā respectively. In fact, the iden-

tity ā = 2ā = 2ā immediately implies that 2G = G = G . One also

has the the analogue of the above Proposition for the Greenā™s operators

G and G . In particular, all the operators d, d , d , Ī“, Ī“ ,ā Ī“ com-

and

mute with the Greenā™s operators. Let us also deļ¬ne dc = ā’1(d ā’ d )

(thus dc is a real operator) and formulate the important

LEMMA 3.9.2 Let Ī· ā Ap,q (M ) be d-exact, then Ī· = ddc Āµ for some

Āµ ā Apā’1,qā’1 (M ).

Proof. First, we notice that the d-exactness of Ī· implies that HĪ· = 0,

therefore, the above Proposition implies that

Ī· = ā G Ī· = d Ī“ G Ī· + Ī“ d G Ī· = d Ī“ G Ī·,

since d G = G d and d-exactness of Ī· imples that d Ī· = d Ī· = 0.

Thus we see that Ī· is d -exact and similarly we can deduce that

Ī· = ā G Ī· = (d Ī“ + Ī“ d )G Ī· = d Ī“ G Ī·.

So we have the following explicit expession:

Ī· = d Ī“ G d Ī“ G Ī· = d d (ā’Ī“ Ī“ G G Ī·),

since Ī“ and ā anti-commute by Corollary 1.16?. Finally, we notice

d

that ddc = 2 ā’1d d .

As a consequence of the above result one has

134 CHAPTER 3. HODGE THEORY

LEMMA 3.9.3 (ddc -Lemma) Let Ī± be a diļ¬erential form on M such

that dc Ī± = 0. If Ī± is d-exact, then Ī± = ddc Ī² for some form Ī².

We leave the details of the proof to the reader.

Further we shall follow the work of Deligne-Morgan-Griļ¬ths-Sullivan

[17]. All the algebras we consider are deļ¬ned over the ļ¬eld of complex

numbers, although many results can be easily generalized to other ļ¬elds.

DEFINITION 3.9.4 A diļ¬erential graded algebra (DGA) is a graded

algebra (over C)

Ai

A=

iā„0

endowed with a diļ¬erential d : Ai ā’ Ai+1 such that

(1) A is (graded) commutative

x Ā· y = (ā’1)ij y Ā· x, x ā Ai , y ā Aj ,

(2) d is a derivation, so that the Leibnitz rule holds:

d(x Ā· y) = dx Ā· y + (ā’1)i x Ā· dy, x ā Ai ,

(3) A is a complex:

d2 = 0.

In a situation like this the cohomology

Ker(d : Ai ā’ Ai+1 )

ā— i i

H (A) = H (A), H (A) =

Im(d : Aiā’1 ā’ Ai )

iā„0

is an algebra itself and we will always assume that dim H ā— (A) < ā.

Here are two important examples of DGAs:

1. The de Rham complex Aā¢ (M ) of diļ¬erential forms on a manifold

M.

2. The cohomology rings H ā— (M, C) of a manifold M with the trivial

diļ¬erential d = 0.

A map between two DGAs A and B is an algebra homomorphism

f : A ā’ B preserving gradings and diļ¬erentials. Such a map induces

an algebra map on cohomology:

f ā— : H ā— (A) ā’ H ā— (B).

3.9. FORMALITY OF COHOMOLOGY 135

Example. Let us take the Dolbeault complex (which is a DGA)

on a compact KĀØhler manifold M of dimension n:

a

d d d

A0,0 (M ) ā’ A0,1 (M ) ā’ Ā· Ā· Ā· ā’ A0,n (M ).

In this situation each cohomology class [Ī±] ā H 0,i (M ) has unique rep-

resentative Ī± ā ā„¦i (M ) - the space of anti-holomorphic forms of degree

i. Since ā„¦ā¢ (M ) is a DGA with zero diļ¬erential we can deļ¬ne a map of

DGAs

ā„¦ā¢ (M ) ā’ A0,ā¢ (M ), Ī± ā’ Ī±,

which induces an isomorphism on cohomology.

The following result was proved in [17] using ddc -Lemma.

PROPOSITION 3.9.5 Let M be a compact KĀØhler manifold and let

a

also {Ac (M ), d} be the sub-DGA of {A (M ), d} consisting of dc -closed

ā— ā—

diļ¬erential forms. Then both DGA maps

i p

{Aā— (M ), d} ā’ {Aā— (M ), d} and {Aā— (M ), d} ā’ {H ā— (M, C), d = 0}

c c

induce isomorphisms on cohomology.

In a situation like this when we have two DGA maps A1 ā A2 ā’ A3

which both induce isomorphisms on cohomology we say that DGAs A1

and A3 are quasi-isomorphic.

Proof. Let {Hc (M ), dind } denote the quotient complex Aā— (M )/dc Aā— (M ).

ā—

c

Then we have the DGA map

Ļ : {Aā— (M ), d} ā’ {Hc (M ), dind }.

ā—

c

If y ā Aā— (M ), i.e. dc y = 0, then by applying the ddc -Lemma we have

c

c

dy = dd z for some diļ¬erential form z. This immediately tells us that

dy lies in the image of the operator dc . Thus, applying Ļ we see that

the induced diļ¬erential dind Ļ(y) = Ļ(dy) = 0. Therefore the induced

diļ¬erential dind on Hc (M ) is zero. For a compact KĀØhler manifold the

ā—

a

DGA Hc (M ) is certainly just the cohomology algebra H ā— (M ) with zero

ā—

diļ¬erential, and, moreover, Ļ coincides with p.

136 CHAPTER 3. HODGE THEORY

To show that p induces an isomorphism on cohomology we need

to establish that the induced map on cohomology pā— is one-to-one and

onto. Let y ā Aā— (M ), dy = 0 satisfy p(y) = 0. It implies that y = dc z

c

for some diļ¬erential form z. By the ddc -Lemma then y = ddc t which

means that y is exact in Aā— (M ) and thus pā— is one-to-one. Conversely,

c

assume that p(y) ā Hc (M ); then y itself is dc -closed and by the ddc -

ā—

Lemma dy = ddc z for some diļ¬erential form z. We let t = y ā’ dc z,

which gives us dt = dy ā’ ddc z = 0 and therefore p(y) and p(t) deļ¬ne

ā—

the same class in Hc (M )

To complete the proof we need to show that iā— is isomorphism on

cohomology. Let [a] be a cohomology class in H ā— (M ) represented by

a d-closed diļ¬erential form a. The form dc a satisļ¬es the hypothesis of

the ddc -Lemma. Thus dc a = ddc b for some diļ¬erential form b. Now

we create a new form by letting Īŗ = a + db. Further, we have dc Īŗ =

dc a + dc db = dc a ā’ ddc b = 0. Therefore, Īŗ is a dc -closed diļ¬erential

form deļ¬ning the same cohomology class [a]. This shows that iā— is

onto. Let now b be a d-closed form in Aā— (M ) which is d-exact in

c

ā—

A (M ), i.e. i(b) = da for some diļ¬erential form a. Then one sees that

the form i(b) completely satisļ¬es the assumptions of the ddc -Lemma,

since dc (i(b)) = i(dc (b)) = 0 and i(b) = da. Thus i(b) = d(dc Īŗ), but

then (dc Īŗ) ā Aā— (M ), since dc (dc Īŗ) = 0. This implies that b is d-exact

c

in Ac (M ) and thus iā— is one-to-one.

ā—

We notice that the way the proof was contrived, the above Propo-

sition remains valid (replacing H ā— (M, C) by Hc (M )) not only for a

ā—

compact KĀØhler manifold M , but also for any compact complex man-

a

ifold M for which the ddc -Lemma holds. A large class of examples of

such manifolds is given by so-called Moishezon manifolds. These man-

ifolds have the property that after a series of blow-ups they become

KĀØhler.

a

We say that A is formal if it is quasi-isomorphic to its cohomology

algebra H ā— (A). Now the main theorem of [17] reads as

THEOREM 3.9.6 (Deligne-Griļ¬ths-Morgan-Sullivan) Let M

be a compact complex manifold for which the ddc -Lemma holds (e.g. a

KĀØhler manifold). Then the DGA Aā— (M ) is formal.

a

Ć¦

Chapter 4

Complex manifolds and

algebraic varieties

Everything worth knowing

can not be taught in a

ńņš. 22 |