Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 267
Zariski sets are dense in X for the usual topology. The restriction of the Zariski
topology on X to Y is also called the Zariski topology of Y .WhenY is irreducible,
the non-empty Zariski open subsets are also dense in Y but this is not the case for
reducible analytic sets.
There is a minimal analytic subset sing(Y ) in X such that Y \ sing(Y ) is a
(smooth) complex submanifold of X \ sing(Y ), i.e. a complex manifold which is
closed and without boundary in X \ sing(Y ). The analytic set sing(Y) is the singular
part of Y .Theregular part of Y is denoted by reg(Y); it is equal to Y \ sing(Y ).The
manifold reg(Y ) is not necessarily irreducible; it may have several components. We
call dimension of Y ,dim(Y ), the maximum of the dimensions of these components;
the codimension codim(Y ) of Y in X is the integer k − dim(Y ). We say that Y is a
proper analytic set of X if it has positive codimension. When all the components of
Y have the same dimension, we say that Y is of pure dimension or of pure codimension.
Whensing(Y ) is non-empty, its dimension is always strictly smaller than the
dimension of Y. We can again decompose sing(Y) into regular and singular parts.
The procedure can be repeated less than k times and gives a stratification of Y into
disjoint complex manifolds. Note that Y is irreducible if and only if reg(Y) is a
connected manifold. The following result is due to Wirtinger.
Theorem A.2 (Wirtinger). Let Y be analytic set of pure dimension p of a Hermitian
manifold (X,ω). Then the 2p-dimensional volume of Y on a Borel set K is equal to
volume(Y ∩ K)= 1
�
ω
p! reg(Y )∩K
p .
Here, the volume is with respect to the Riemannian metric induced by ω.
Let Dk denote the unit polydisc {|z1| < 1,...,|zk| < 1} in C k . The following
result describes the local structure of analytic sets.
Theorem A.3. Let Y be an analytic set of pure dimension p of X. Let a be a point
of Y. Then there is a holomorphic chart U of X, bi-holomorphic to Dk, with local
coordinates z =(z1,...,zk), such that z = 0 at a, U is given by {|z1| < 1,...,|zk| < 1}
and the projection π : U → Dp, defined by π(z) :=(z1,...,zp), is proper on Y ∩U.
In this case, there is a proper analytic subset S of Dp such that π : Y ∩U \π −1 (S) →
Dp \ S is a finite covering and the singularities of Y are contained in π −1 (S).
Recall that a holomorphic map τ : X1 → X2 between complex manifolds of the
same dimension is a covering of degree d if each point of X2 admits a neighbourhood
V such that τ −1 (V) is a disjoint union of d open sets, each of which is sent
bi-holomorphically to V by τ. Observe the previous theorem also implies that the
fibers of π : Y ∩U → Dp are finite and contain at most d points if d is the degree of
the covering. We can reduce U in order to have that a is the unique point in the fiber
π −1 (0) ∩Y. The degree d of the covering depends on the choice of coordinates and
the smallest integer d obtained in this way is called the multiplicity of Y at a and
is denoted by mult(Y,a). We will see that mult(Y,a) is the Lelong number at a of
the positive closed current associated to Y. In other words, if Br denotes the ball of