Discrete Holomorphic Local Dynamical Systems

160 Marco Brunella
rational
quasi−fibrations
parabolic
foliations
hyperbolic
foliations
Let us discuss the classical case of fibrations.
foliations by rational curves
foliations whose canonical bundle
is not pseudoeffective
Example 8.6. Suppose that F is a fibration over some base B, i.e. there exists a
holomorphic map f : X → B whose generic fiber is a leaf of F (but there may be
singular fibers, and even some higher dimensional fibers). Let g be the genus of a
generic fiber, and suppose that g ≥ 1. The relative canonical bundle of f is defined as
Kf = KX ⊗ f ∗ (K −1
B ).
It is related to the canonical bundle KF of F by the relation
Kf = KF ⊗ OX(D)
where D is an effective divisor which takes into account the possible ramifications
of f along nongeneric fibers. Indeed, by adjunction along the leaves, we have KX =
KF ⊗ N∗ F ,whereN∗ F denotes the determinant conormal bundle of F .Ifω is a
local generator of KB,thenf∗ (ω) is a local section of N∗ F which vanishes along the
ramification divisor D of f , hence f ∗ (KB) =N∗ F ⊗ OX (−D), whence the relation
above.
Because F is not a foliation by rational curves, we have, by the Theorems above,
that KF is pseudoeffective, and therefore also Kf is pseudoeffective. In particular,
f∗(KF ) and f∗(Kf ) are “pseudoeffective sheaves” on B, in the sense that their degrees
with respect to Kähler metrics on B are nonnegative. This must be compared
with Arakelov’s positivity theorem [Ara] [BPV, Ch. III]. But, as in Arakelov’s results,
something more can be said. Suppose that B is a curve (or restrict the fibration
f over some curve in B) and let us distinguish between the hyperbolic and the
parabolic case.
• g ≥ 2. Then the pseudoeffectivity of KF is realized by the leafwise Poincaré
metric (Theorem 6.4). A subtle computation [Br2, Br1] shows that this leafwise
(or fiberwise) Poincaré metric has a strictly plurisubharmonic variation, unless
the fibration is isotrivial. This means that if f is not isotrivial then the degree of
f∗(KF ) (and, a fortiori, the degree of f∗(Kf ))isstrictly positive.