Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 265
Let H p,q (X,C), 0≤ p,q ≤ k, denote the subspace of H p+q (X,C) generated
by the classes of closed (p,q)-forms. We call H p,q (X,C) the Hodge cohomology
group. Hodge theory shows that
H l (X,C)= �
p+q=l
H p,q (X,C) and H q,p (X,C)=H p,q (X,C).
This, together with the Poincaré duality, induces a canonical isomorphism between
H p,q (X,C) and the dual space of H k−p,k−q (X,C). Defineforp = q
We have
H p,p (X,R) := H p,p (X,C) ∩ H 2p (X,R).
H p,p (X,C)=H p,p (X,R) ⊗R C.
Recall that the Dolbeault cohomology group H p,q
(X) is the quotient of the space
∂
of ∂-closed (p,q)-forms by the subspace of ∂-exact (p,q)-forms. Observe that a
(p,q)-form is d-closed if and only if it is ∂-closed and ∂ -closed. Therefore, there
is a natural morphism between the Hodge and the Dolbeault cohomology groups.
Hodge theory asserts that this is in fact an isomorphism: we have
H p,q (X,C) � H p,q
(X).
∂
The result is a consequence of the following theorem, the so-called ddc-lemma,see e.g. [DEM, VO].
Theorem A.1. Let ϕ be a smooth d-closed (p,q)-form on X. Then ϕ is ddc-exact if
and only if it is d-exact (or ∂-exact or ∂-exact).
The projective space Pk admits a Kähler form ωFS, called the Fubini-Study form.
It is defined on the chart Ui by
ωFS := dd c �
k �
�
log � z �
j �
� 2
�
.
In other words, if π : Ck+1 \{0} →Pk is the canonical projection, then ωFS is
defined by
π ∗ (ωFS) := dd c �
k
log
�
�zi| 2
�
.
One can check that ω k FS is a probability measure on Pk . The cohomology groups
of Pk are very simple. We have H p,q (Pk ,C) =0forp�=q and H p,p (Pk ,C) � C.
The groups H p,p (Pk ,R) and H p,p (Pk ,C) are generated by the class of ω p
FS . Submanifolds
of Pk are Kähler, as submanifolds of a Kähler manifold. Chow’s theorem
says that such a manifold is algebraic, i.e. it is the set of common zeros of a finite
family of homogeneous polynomials in z. A compact manifold is projective if it is
bi-holomorphic to a submanifold of a projective space. Their cohomology groups
are in general very rich and difficult to describe.
∑
j=0
zi
∑
i=0