Discrete Holomorphic Local Dynamical Systems

152 Marco Brunella
pseudoconvex neighbourhoods of the null section. Recall that an open subset W of
a complex manifold E is locally pseudoconvex in E if for every w ∈ ∂W there exists
a neighbourhood Uw ⊂ E of w such that W ∩Uw is Stein.
Lemma 7.4. Let X be a compact connected complex manifold and let L be a line
bundle on X. The following two properties are equivalent:
(i) L is pseudoeffective;
(ii) in the total space E ∗ of the dual line bundle L ∗ there exists a neighbourhood
W �= E ∗ of the null section Σ ∗ which is locally pseudoconvex in E ∗ .
Proof. The implication (i) ⇒ (ii) is quite evident. If h is a (singular) hermitian metric
on L with positive curvature [Dem], then in a local trivialization E|Uj � Uj × C
the unit ball is expressed by {(z,t) ||t| < e h j(z) },whereh j : Uj → [−∞,+∞) is the
plurisubharmonic weight of h. In the dual local trivialization E ∗ |Uj � Uj × C, the
unit ball of the dual metric is expressed by {(z,s) ||s| < e −h j(z) }. The plurisubharmonicity
of h j gives (and is equivalent to) the Steinness of such an open subset of
Uj × C (recall Hartogs Theorem on Hartogs Tubes mentioned in Section 2). Hence
we get (ii), with W equal to the unit ball in E ∗ .
The implication (ii) ⇒ (i) is not more difficult. Let W ⊂ E ∗ be as in (ii). On E ∗
we have a natural S 1 -action, which fixes Σ ∗ and rotates each fiber. For every ϑ ∈ S 1 ,
let Wϑ be the image of W by the action of ϑ.Then
W ′ = ∩ ϑ∈S 1Wϑ
is still a nontrivial locally pseudoconvex neighbourhood of Σ ∗ , for local pseudoconvexity
is stable by intersections. For every z ∈ X, W ′ intersects the fiber E ∗ z along
an open subset which is S1-invariant, a connected component of which is therefore
adiscW0 z centered at the origin (possibly W 0
z = E∗ z for certain z, but not for all);
the other components are annuli around the origin. Using the local pseudoconvexity
of W ′ , i.e. its Steinness in local trivializations E∗ |Uj � Uj × C, it is easy to see that
these annuli and discs cannot merge when z moves in X.Inotherwords,
W ′′ = ∪z∈XW 0
z
is a connected component of W ′ , and of course it is still a nontrivial pseudoconvex
neighbourhood of Σ ∗ .WemayuseW ′′ as unit ball for a metric on L ∗ .Asinthefirst
part of the proof, the corresponding dual metric on L has positive curvature, in the
sense of currents. ⊓⊔
Consider now, in the space UΣ × Y, the graph Γf of the meromorphic map
f : U 0 Σ = UΣ \ Σ ��� Y. By definition of meromorphicity, Γf is an irreducible analytic
subset of U 0 Σ ×Y ⊂ UΣ ×Y, whose projection to U 0 Σ is proper and generically
bijective. It may be singular, and in that case we replace it by a resolution of its
singularities, still denoted by Γf . The (new) projection
π : Γf → U 0 Σ