Discrete Holomorphic Local Dynamical Systems

110 Marco Brunella
We put this inequality into the expression of ∂ 2 λ
∂z∂ ¯z (z0) derived above from Green
formula, and then we apply Stokes theorem. We find
∂ 2λ ∂z∂ ¯z (z0) ≤− 2
� �
�
�
∂
π �
Dz0 2g ∂w∂ ¯z (z0,w)
�2
�
�
� idw ∧ d ¯w ≤ 0
from which we see that λ is superharmonic. ⊓⊔
A similar result can be proved, by the same proof, even when we drop the simply
connectedness hypothesis on the fibers, for instance when the fibers of U0 are annuli
instead of discs; however, the result is that the Bergman fiberwise metric, and not
the Poincaré one, has a plurisubharmonic variation. This is because on a multiply
connected curve the Green function is more directly related to the Bergman metric
[Ya3]. The case of the Poincaré metric is done in [Kiz], by a covering argument. The
general case of Theorem 2.1 requires also to understand what happens when ∂U0
is still pseudoconvex but no more transverse to the fibers, so that U0 is no more a
differentiably trivial family of curves. This is rather delicate, and it is done in [Ya1].
Then Theorem 2.1 is proved by an exhaustion argument.
2.2 Parabolic Fibrations
Theorem 2.1, as stated, is rather empty when all the fibers are isomorphic to C.
However, in that case Nishino proved that if U is Stein then it is isomorphic to
D n × C [Nis, II]. A refinement of this was found in [Ya2].
As before, we consider a fibration P : U → D n and we do not assume that U is
Stein. We suppose that there exists an embedding j : D n × D → U such that P ◦ j
coincides with the projection from D n × D to D n (this can always be done, up to
restricting the base). For every ε ∈ [0,1),weset
Uε = U \ j(D n × D(ε))
with D(ε)={z ∈ C| |z|≤ε}, and we denote by
Pε : Uε → D n
the restriction of P. Thus, the fibers of Pε are obtained from those of P by removing
a closed disc (if ε > 0) or a point (if ε = 0).
Theorem 2.3. [Nis, Ya2, II] Suppose that:
(i) for every z ∈ Dn , the fiber P−1 (z) is isomorphic to C;
(ii) for every ε > 0 the fiberwise Poincaré metriconUε Pε
→ Dn has a plurisubharmonic
variation.
Then U is isomorphic to a product:
U � D n × C.