Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 151
for every n. By assumption (B), the support of χn ◦ Φ ∗ intersects V \ U along a
compact subset. Moreover, Φ ∗ is identically zero on U. Thus, the integrand above
has compact support in V,aswellas(χn ◦Φ ∗ )·d c Φ ∗ ∧η. Hence, by Stokes formula,
�
In = − d(χn ◦ Φ
V
∗ ) ∧ d c Φ ∗ �
∧ η = − (χ
V
′ n ◦ Φ ∗ ) · dΦ ∗ ∧ d c Φ ∗ ∧ η.
Now, dΦ ∗ ∧d c Φ ∗ is a positive current, and its product with η is a positive measure.
From χ ′ n ≤ 0 we obtain In ≥ 0, for every n. ⊓⊔
This inequality completes the proof of the theorem. ⊓⊔
7.2 Extension of Meromorphic Maps
As in [Din, §6], we shall use the volume estimate of Theorem 7.1 to get an extension
theorem for certain meromorphic maps into Kähler manifolds.
Consider the following situation. It is given a compact connected Kähler manifold
X, ofdimensionn, and a line bundle L on X. Denote by E the total space of
L, and by Σ ⊂ E the graph of the null section of L. LetUΣ ⊂ E be a connected
(tubular) neighbourhood of Σ, andletY be another compact Kähler manifold, of
dimension m.
Theorem 7.3. [Br5] Suppose that L is not pseudoeffective. Then any meromorphic
map
f : UΣ \ Σ ��� Y
extends to a meromorphic map
¯f : UΣ ��� Y.
Before the proof, let us make a link with [BDP]. In the special case where X is
projective, and not only Kähler, the non pseudoeffectivity of L translates into the
existence of a covering family of curves {Ct}t∈B on X such that L|Ct has negative
degree for every t ∈ B [BDP]. This means that the normal bundle of Σ in E has
negative degree on every Ct ⊂ Σ � X. Hence the restriction of E over Ct is a surface
Et which contains a compact curve Σt whose selfintersection is negative, and thus
contractible to a normal singularity. By known results [Siu] [Iv1], every meromorphic
map from Ut \ Σt (Ut being a neighbourhood of Σt in Et) into a compact Kähler
manifold can be meromorphically extended to Ut. Because the curves Ct cover the
full X, this is sufficient to extends from UΣ \ Σ to UΣ .
Of course, if X is only Kähler then such a covering family of curves could not
exist, and we need a more global approach, which avoids any restriction to curves.
Even in the projective case, this seems a more natural approach than evoking [BDP].
Proof. We begin with a simple criterion for pseudoeffectivity, analogous to the well
known fact that a line bundle is ample if and only if its dual bundle has strongly