Discrete Holomorphic Local Dynamical Systems

84 Eric Bedford
(Exercise: The most obvious first attempt, the one point compactification Ĉ 2 of C 2 ,
is not a complex manifold.) We note, too, that we do not discuss are the complex
2-tori or the K3 surfaces, which are the other possibilities for projective surfaces
with automorphisms for which λ ( f ) > 1(see[C1]). In fact, rational surface automorphisms
are more abundant than these other two cases, so we will be drawing
from a rich family of dynamical systems.
In §1 we used the dynamical classification of PolyAut(C 2 ) which shows that the
Hénon diffeomorphisms represent the conjugacy classes with positive entropy. On
the other hand, it is not easy to determine the set of all rational surfaces that admit
nontrivial automorphisms; and given such a surface, it is not easy to determine its
automorphisms. In other words, an analogous dynamical classification of rational
surface automorphisms is not yet known.
Here we focus on the surfaces that are obtained from P 2 by blow-ups; this
focus is justified by a Theorem of Nagata which is stated in §2.7. However,
we note that a surface constructed from P 2 by making “generic” blowups will
not have any automorphisms except the identity (see [H, K]). Our starting place
will be complex projective space P n ,whichisC n+1 − 0, modulo the equivalence
(x0,...,xn) ∼ (λ x0,...,λ xn) for any λ ∈ C with λ �= 0. We write [x0 : ···: xn] (square
brackets to denote homogeneous coordinates) for the equivalence class. It is classical
that Aut(P n )=PGL(n + 1,C), andAut(P 1 × P 1 ) is the group generated by
PGL(2,C) × PGL(2,C), together with the map τ(x,y)=(y,x).
The blow-up of a manifold X at a point p ∈ X is a manifold ˜X, together with a
projection π : ˜X → X such that the exceptional fiber E := π −1 p is equivalent to P 1 ,
and π : ˜X −E → X − p is biholomorphic. A concrete presentation of this is the space
Γ = {(x,y);[ξ : η] ∈ C 2 × P 1 : xη = yξ }
together with the projection π to the first coordinate, so we define the pair (Γ ,π)
to be the blow-up of C 2 at 0. This construction is essentially local at the center of
blowup. Thus we may use this construction of Γ to define the blowup of X at the
point p.
Observe that π −1 : C 2 − 0 → Γ is the tautological map π −1 (x,y)=(x,y);[x : y].
We may also write π −1 (x,y) =(x,y);[1 :y/x] =(x,y);[x/y :1], andwemayuse
these two representations to define local coordinate charts U ′ = C 2 s,η and U ′′ = C 2 ξ ,t ,
where the coordinates are defined via the form that the projection π takes:
π ′ : (s,η) ↦→ (s,sη)=(x,y), π ′′ : (ξ ,t) ↦→ (ξt,t)=(x,y).
It follows that U ′ ∪U ′′ = Γ ,andE ′ := E ∩U ′ = {s = 0} is equivalent to C.
We may pull the 2-form dx∧ dy back to Γ ;intheU ′ coordinate chart, for instance,
we have (π ′ ) ∗ dx∧ dy = ds∧ d(sη)=sds∧ dη.
We will use the following notational convention: if C is a curve in X,thenC will
also denote the strict transform which is the curve in ˜X which is obtained by taking
the closure of π −1 (C − p) inside ˜X. In the blowup, the strict transform of the x-axis
{y = 0} inside Γ is given by {η = 0}. Thus we use the (s,η) coordinate system if
we want to work in a neighborhood of the x-axis, and we use the (ξ ,t) coordinate
system if we want to work in a neighborhood of the y-axis.