Discrete Holomorphic Local Dynamical Systems

46 Marco Abate
and dimE = 1; but this part of the argument works for any n ≥ 2 (even when E has
singularities, and it can also be adapted to non-tangential germs).
Since dimE = 1 = rkNE, the restriction of the canonical morphism Xf to N ⊗ν f
Eo is an isomorphism between N ⊗ν f
Eo and TEo .Thenin[ABT1] we showed that it is
possible to define a holomorphic connection ∇ on NEo by setting
∇u(s)=π([X f (ũ), ˜s]|S), (44)
where: s is a local section of NEo; u ∈ TEo ; π : TM|Eo → NEo is the canonical
projection; ˜s is any local section of TM|Eo such that π( ˜s|S o)=s; ũ is any local
section of TM⊗ν �
f such that Xf π(ũ|E o)� = u; andXfis locally given by (43).
In a chart (U,z) adapted to E, a local generator of NEo is ∂1 = π(∂/∂z1), a local
generator of N ⊗ν f
E o
therefore
is ∂ ⊗ν f
1
= ∂1 ⊗···⊗∂1, and we have
Xf (∂ ⊗ν f
1
∇ ∂/∂z2 ∂1 = − 1
∂
)=g2|U∩E
∂z2
�
�
∂g1 �
g2 ∂z1
�
U∩E
In particular, ∇ is a meromorphic connection on NE, with poles in the singular points
of f .
Definition 6.28. The index ιp( f ,E) of f along E at a point p ∈ E is by definition
the opposite of the residue at p of the connection ∇ divided by ν f :
ιp( f ,E)=− 1
Resp(∇).
ν f
;
∂1.
In particular, ιp( f ,E)=0ifp is not a singular point of f .
Remark 6.29. If [v] is a non-degenerate characteristic direction of a non-dicritical
fo ∈ End(C 2 ,O) with non-zero director α ∈ C ∗ , then it is not difficult to check that
ι [v]( f ,E)= 1
α ,
where f is the lift of fo to the blow-up of the origin.
Then in [A2] we proved the following index theorem (see [Br1, BrT, ABT1,
ABT2] for multidimensional versions and far reaching generalizations):
Theorem 6.30 (Abate, Bracci, Tovena, 2004 [A2], [ABT1]). Let E be a compact
Riemann surface inside a 2-dimensional complex manifold M. Take f ∈ End(M,E),
and assume that f is tangential to E. Then
∑ ιq( f ,E)=c1(NE),
q∈E
where c1(NE) is the first Chern class of the normal bundle NE of E in M.