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3.8 Hodge decomposition and sheaf coho-
mology
Now let us give a couple of di¬erent interpretations of the groups
H p,q (M ) arising in the Hodge decomposition

H j (M, C) = •p+q=j H p,q (M )

of cohomology of a compact K¨hler 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 di¬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 ) ’ Ap,1 (M ) ’ · · · ’ Ap,n (M ),

where n = dimC M and Ap,q (M ) is the space of complex-valued di¬er-
ential forms on M of type (p, q).
PROPOSITION 3.8.1 The degree q cohomology of this Dolbeault
complex identi¬es with the space H p,q .
For instance, we have
PROPOSITION 3.8.2 For ± ∈ Ap,0 (M ), ± is holomorphic if and
only if d ± = 0.

COROLLARY 3.8.3

Ker[d : Ap,0 (M ) ’ Ap,1 (M )] = „¦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 ∆ . We know
that ∆ = ∆/2, hence the spaces of di¬erential forms ± of type (p, q)
such that ∆ ± = 0 and ∆± = 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, „¦p ), where „¦p denotes
the sheaf of holomorphic p-forms.
ˇ ˇ
In terms of Cech cohomology H p,q (M ) = H q (M, „¦p ).

The equivalence of those descriptions follows from the theorem which
identi¬es sheaf cohomology with coe¬cients in holomorphic sections of
ˇ
a vector bundle with Cech cohomology with coe¬cients in this bundle
for all reasonable manifolds (e.g. compact).


3.9 Formality of cohomology of compact
K¨hler manifolds
a
In this section we deal with a compact K¨hler manifold M unless oth-
a
erwise stated. By Ap,q (M ) we understand the space of complex-valued
di¬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 ) ’ Hp,q (M ). Since all the cohomology
groups of M are ¬nite-dimenional, H is well-de¬ned. We also recall
that the space Ap,q (M ) is the direct sum of Hp,q (M ) and ∆(Ap,q (M )).
p,q
Let H⊥ (M ) stand for the orthogonal complement of the space of har-
monic forms of type (p, q) in Ap,q (M ). It identi¬es then with the image
of Ap,q (M ) under the action of ∆. It also follows that the operator ∆
p,q
restricted to H⊥ (M ) is invertible.
Let us de¬ne the Green™s operator G : Ap,q (M ) ’ Ap,q (M ) as a
composition
G = (∆|H⊥ (M ) )’1 —¦ (Id ’ H),
p,q
3.9. FORMALITY OF COHOMOLOGY 133

where Id is the identity operator. The basic properties of the Green™s
operator G are summarized in
PROPOSITION 3.9.1 The operators G and H commute with every
linear map T : Ap,q (M ) ’ Ap,q (M ) which commutes with ∆. Besides,
one has:
H∆ = ∆H = 0, GH = HG = 0, Id = H + ∆G = H + G∆.
We leave the proof of this Proposition as an easy exercise to the reader.
One of the other properties of G that we brie¬‚y mention is that the
operator G is continuous (in the natural Fr´chet topology on Ap,q (M )).
e
Analogously one can de¬ne the Green™s operators G and G associ-
ated to the Laplace operators ∆ and ∆ respectively. In fact, the iden-
tity ∆ = 2∆ = 2∆ immediately implies that 2G = G = G . One also
has the the analogue of the above Proposition for the Green™s operators
G and G . In particular, all the operators d, d , d , δ, δ ,√ δ com-
and
mute with the Green™s operators. Let us also de¬ne dc = ’1(d ’ d )
(thus dc is a real operator) and formulate the important

LEMMA 3.9.2 Let · ∈ Ap,q (M ) be d-exact, then · = ddc µ for some
µ ∈ Ap’1,q’1 (M ).

Proof. First, we notice that the d-exactness of · implies that H· = 0,
therefore, the above Proposition implies that
· = ∆ G · = d δ G · + δ d G · = d δ G ·,
since d G = G d and d-exactness of · imples that d · = d · = 0.
Thus we see that · is d -exact and similarly we can deduce that
· = ∆ G · = (d δ + δ d )G · = d δ G ·.
So we have the following explicit expession:
· = d δ G d δ G · = d d (’δ δ G G ·),
since δ and √ anti-commute by Corollary 1.16?. Finally, we notice
d
that ddc = 2 ’1d d .

As a consequence of the above result one has
134 CHAPTER 3. HODGE THEORY

LEMMA 3.9.3 (ddc -Lemma) Let ± be a di¬erential form on M such
that dc ± = 0. If ± is d-exact, then ± = ddc β for some form β.
We leave the details of the proof to the reader.
Further we shall follow the work of Deligne-Morgan-Gri¬ths-Sullivan
[17]. All the algebras we consider are de¬ned over the ¬eld of complex
numbers, although many results can be easily generalized to other ¬elds.

DEFINITION 3.9.4 A di¬erential graded algebra (DGA) is a graded
algebra (over C)
Ai
A=
i≥0

endowed with a di¬erential d : Ai ’ Ai+1 such that
(1) A is (graded) commutative

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

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

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

(3) A is a complex:
d2 = 0.

In a situation like this the cohomology
Ker(d : Ai ’ Ai+1 )
— i i
H (A) = H (A), H (A) =
Im(d : Ai’1 ’ Ai )
i≥0

is an algebra itself and we will always assume that dim H — (A) < ∞.
Here are two important examples of DGAs:
1. The de Rham complex A• (M ) of di¬erential forms on a manifold
M.
2. The cohomology rings H — (M, C) of a manifold M with the trivial
di¬erential d = 0.
A map between two DGAs A and B is an algebra homomorphism
f : A ’ B preserving gradings and di¬erentials. Such a map induces
an algebra map on cohomology:

f — : H — (A) ’ H — (B).
3.9. FORMALITY OF COHOMOLOGY 135


Example. Let us take the Dolbeault complex (which is a DGA)
on a compact K¨hler manifold M of dimension n:
a
d d d
A0,0 (M ) ’ A0,1 (M ) ’ · · · ’ A0,n (M ).

In this situation each cohomology class [±] ∈ H 0,i (M ) has unique rep-
resentative ± ∈ „¦i (M ) - the space of anti-holomorphic forms of degree
i. Since „¦• (M ) is a DGA with zero di¬erential we can de¬ne a map of
DGAs
„¦• (M ) ’ A0,• (M ), ± ’ ±,
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 K¨hler manifold and let
a
also {Ac (M ), d} be the sub-DGA of {A (M ), d} consisting of dc -closed
— —

di¬erential forms. Then both DGA maps
i p
{A— (M ), d} ’ {A— (M ), d} and {A— (M ), d} ’ {H — (M, C), d = 0}
c c

induce isomorphisms on cohomology.

In a situation like this when we have two DGA maps A1 ← A2 ’ 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 A— (M )/dc A— (M ).

c
Then we have the DGA map

ρ : {A— (M ), d} ’ {Hc (M ), dind }.

c

If y ∈ A— (M ), i.e. dc y = 0, then by applying the ddc -Lemma we have
c
c
dy = dd z for some di¬erential form z. This immediately tells us that
dy lies in the image of the operator dc . Thus, applying ρ we see that
the induced di¬erential dind ρ(y) = ρ(dy) = 0. Therefore the induced
di¬erential dind on Hc (M ) is zero. For a compact K¨hler manifold the

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


di¬erential, and, moreover, ρ 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 p— is one-to-one and
onto. Let y ∈ A— (M ), dy = 0 satisfy p(y) = 0. It implies that y = dc z
c
for some di¬erential form z. By the ddc -Lemma then y = ddc t which
means that y is exact in A— (M ) and thus p— is one-to-one. Conversely,
c
assume that p(y) ∈ Hc (M ); then y itself is dc -closed and by the ddc -


Lemma dy = ddc z for some di¬erential form z. We let t = y ’ dc z,
which gives us dt = dy ’ ddc z = 0 and therefore p(y) and p(t) de¬ne

the same class in Hc (M )
To complete the proof we need to show that i— is isomorphism on
cohomology. Let [a] be a cohomology class in H — (M ) represented by
a d-closed di¬erential form a. The form dc a satis¬es the hypothesis of
the ddc -Lemma. Thus dc a = ddc b for some di¬erential form b. Now
we create a new form by letting κ = a + db. Further, we have dc κ =
dc a + dc db = dc a ’ ddc b = 0. Therefore, κ is a dc -closed di¬erential
form de¬ning the same cohomology class [a]. This shows that i— is
onto. Let now b be a d-closed form in A— (M ) which is d-exact in
c

A (M ), i.e. i(b) = da for some di¬erential form a. Then one sees that
the form i(b) completely satis¬es the assumptions of the ddc -Lemma,
since dc (i(b)) = i(dc (b)) = 0 and i(b) = da. Thus i(b) = d(dc κ), but
then (dc κ) ∈ A— (M ), since dc (dc κ) = 0. This implies that b is d-exact
c
in Ac (M ) and thus i— is one-to-one.




We notice that the way the proof was contrived, the above Propo-
sition remains valid (replacing H — (M, C) by Hc (M )) not only for a


compact K¨hler 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
K¨hler.
a
We say that A is formal if it is quasi-isomorphic to its cohomology
algebra H — (A). Now the main theorem of [17] reads as
THEOREM 3.9.6 (Deligne-Gri¬ths-Morgan-Sullivan) Let M
be a compact complex manifold for which the ddc -Lemma holds (e.g. a
K¨hler manifold). Then the DGA A— (M ) is formal.
a
¦
Chapter 4

Complex manifolds and
algebraic varieties

Everything worth knowing
can not be taught in a

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