Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 269
The following desingularization theorem, due to Hironaka, is very useful.
Theorem A.6. Let Z be an analytic space. Then there is a smooth manifold �Z,
possibly reducible, and a holomorphic map π : �Z → Z such that π −1 (reg(Z))
is a dense Zariski open set of �Z and π defines a bi-holomorphic map between
π −1 (reg(Z)) and reg(Z).
When Z is an analytic subset of a manifold X, then one can obtain a map
π : �X → X using a sequence of blow-ups along the singularities of Z. The manifold
�Z is the strict transform of Z by π. The difference with the normalization of Z is
that we do not have the second property in Theorem A.4 but �Z is smooth.
Exercise A.7. Let X be a compact Kähler manifold of dimension k. Show that the
Betti number bl, i.e. the dimension of H l (X,R), isevenifl is odd and does not
vanish if l is even.
Exercise A.8. Let Grass(l,k) denote the Grassmannian, i.e. the set of linear subspaces
of dimension l of C k . Show that Grass(l,k) admits a natural structure of a
projective manifold.
Exercise A.9. Let X be a compact complex manifold of dimension ≥ 2 and
π : �X × X → X × X the blow-up of X × X along the diagonal Δ. LetΠ1,Π2 denote
the natural projections from �X × X onto the two factors X of X × X. Show that
Π1,Π2 and their restrictions to π −1 (Δ) are submersions.
Exercise A.10. Let E be a finite or countable union of proper analytic subsets of a
connected manifold X. Show that X \ E is connected and dense in X for the usual
topology.
Exercise A.11. Let τ : X1 → X2 be a ramified covering of degree n. Letϕbe a
function on X1.Define
τ∗(ϕ)(z) := ∑
w∈τ−1 (z)
ϕ(w),
where the points in τ −1 (z) are counted with multiplicity. If ϕ is upper semicontinuous
or continuous, show that τ∗(ϕ) is upper semi-continuous or continuous
respectively. Show that the result still holds for a general open map τ between
manifolds of the same dimension if ϕ has compact support in X1.
A.2 Positive Currents and p.s.h. Functions
In this paragraph, we introduce positive forms, positive currents and plurisubharmonic
functions on complex manifolds. The concept of positivity and the notion
of plurisubharmonic functions are due to Lelong and Oka. The theory has many
applications in complex algebraic geometry and in dynamics.