Discrete Holomorphic Local Dynamical Systems

148 Marco Brunella
(thanks to the plurisubharmonic variation of the fiberwise Poincaré metriconUT ),
and being also exhaustive we deduce that UT is Stein. Probably, this can be done also
if the fiberwise Poincaré metric is less regular, say C 0 . But when there are parabolic
fibers such a simple argument cannot work, because ψ is no more exhaustive (one
can try perhaps to use a renormalization argument like the one used in the proof of
Theorem 2.3). However, if all the fibers are parabolic then we shall see later that UT
is a product T × C (if T is small), and hence it is Stein.
A related problem concerns the existence on UT of holomorphic functions which
are not constant on the fibers. By Corollary 6.5, KF is pseudoeffective, if F is hyperbolic.
Let us assume a little more, namely that it is effective. Then any nontrivial
section of KF over X can be lifted to UT , giving a holomorphic section of the relative
canonical bundle of the fibration. As in Lemmata 2.5 and 2.6, this section can
be integrated along the (simply connected and pointed) fibers, giving a holomorphic
function on UT not constant on generic fibers.
7 Extension of Meromorphic Maps from Line Bundles
In order to generalize Corollary 6.5 to cover (most) parabolic foliations, we need
an extension theorem for certain meromorphic maps. This is done in the present
Section, following [Br5].
7.1 Volume Estimates
Let us firstly recall some results of Dingoyan [Din], in a slightly simplified form due
to our future use.
Let V be a connected complex manifold, of dimension n, andletω be a smooth
closed semipositive (1,1)-form on V (e.g., the pull-back of a Kähler form by some
holomorphic map from V). Let U ⊂ V be an open subset, with boundary ∂U com-
pact in V . Suppose that the mass of ω n on U is finite: �
U ωn < +∞. We look for
some condition ensuring that also the mass on V is finite: �
V ωn < +∞. Inother
words, we look for the boundedness of the ω n -volume of the ends V \U.
Set
Pω(V,U)={ϕ : V → [−∞,+∞) u.s.c. | dd c ϕ + ω ≥ 0, ϕ|U ≤ 0}
where u.s.c. means upper semicontinuous, and the first inequality is in the sense of
currents. This first inequality defines the so-called ω-plurisubharmonic functions.
Note that locally the space of ω-plurisubharmonic functions can be identified with
a translation of the space of the usual plurisubharmonic functions: locally the form
ω admits a smooth potential φ (ω = dd c φ), and so ϕ is ω-plurisubharmonic
is and only if ϕ + φ is plurisubharmonic. In this way, most local problems
on ω-plurisubharmonic functions can be reduced to more familiar problems on
plurisubharmonic functions.