Discrete Holomorphic Local Dynamical Systems

122 Marco Brunella
which sends the discrete subset Lp \ L 0 p into Sing(F ). Note that ip mayfailtobe
immersive at those points. Moreover, it may happen that two different points of
Lp \ L 0 p are sent by ip to the same singular point of F (see Example 4.4 below). In
spite of this, we shall sometimes identify Lp with its image in X. For instance, to say
that a map f : Z → X “has values into Lp” shall mean that f factorizes through ip.
Remark that we have not defined, and shall not define, leaves Lp through p ∈
Sing(F ): a leaf may pass through Sing(F ), but its basepoint must be chosen outside
Sing(F ).
Let us see two examples.
Example 4.3. Take a compact Kähler surface S foliated by an elliptic fibration π :
S → C, andletc0∈Cbe such that the fiber F0 = π−1 (c0) is of Kodaira type II
[BPV, V.7], i.e. a rational curve with a cusp q. Ifp ∈ F0, p �= q, then the leaf L0 p is
equal to F0 \{q}�C. This leaf has a parabolic end with trivial holonomy, which is
not a vanishing end. Indeed, this end can be compactified to a cuspidal disc, which
however cannot be meromorphically deformed as a disc to nearby leaves, because
nearby leaves have positive genus close to q. Hence Lp = L0 p .
Let now �S → S be the composition of three blow-ups which transforms F0 into
a tree of four smooth rational curves �F0 = G1 + G2 + G3 + G6 of respective selfintersections
−1,−2,−3,−6[BPV, V.10]. Let �π : �S → C be the new elliptic fibration/foliation.
Set p j = G1 ∩ G j, j = 2,3,6. If p ∈ G1 is different from those three
points, then L0 p = G1 \{p2, p3, p6}. The parabolic end of L0 p corresponding to p2
(resp. p3, p6) has holonomy of order 2 (resp. 3, 6). This time, this is a vanishing
end: a disc D in G1 through p2 (resp. p3, p6) ramified at order 2 (resp. 3, 6) can be
deformed to nearby leaves as discs close to 2D + G2 (resp. 3D + G3,6D + G6), and
also the “meromorphic immersion” condition can be easily respected. Thus Lp is
isomorphic to the orbifold “P with three points of multiplicity 2, 3, 6”. Note that the
universal covering (in orbifold sense) of Lp is isomorphic to C, and the holonomy
covering (defined below) is a smooth elliptic curve.
Finally, if p ∈ G j, p �= p j, j = 2,3,6, then L0 p has a parabolic end with trivial
holonomy, which is not a vanishing end, and so Lp = L0 p � C.
A more systematic analysis of the surface case, from a slightly different point of
view, can be found in [Br1].
Example 4.4. Take a projective threefold M containing a smooth rational curve C
with normal bundle NC = O(−1) ⊕ O(−1). Take a foliation F on M, nonsingular
around C, such that: (i) for every p ∈ C, TpF is different from TpC; (ii) TF has
degree -1 on C. It is easy to see that there are a lot of foliations on M satisfying
these two requirements. Note that, on a neighbourhood of C, we can glue together
the local leaves (discs) of F through C, and obtain a smooth surface S containing
C; condition (ii) means that the selfintersection of C in S is equal to -1.
We now perform a flop of M along C. That is, we firstly blow-up M along C, obtaining
a threefold �M containing an exceptional divisor D naturally P-fibered over C.
Because NC = O(−1) ⊕ O(−1), this divisor D is in fact isomorphic to P × P, hence
it admits a second P-fibration, transverse to the first one. Each fibre of this second
fibration can be blow-down to a point (Moishezon’s criterion [Moi]), and the result