Discrete Holomorphic Local Dynamical Systems

114 Marco Brunella
lifted to �Lt). Then we can repeat the argument above: the fiberwise iterated integral
of ( �ω1, �ω2) is a holomorphic function on UT which separates p and q. ⊓⊔
Having established that UT is a holomorphically separable Riemann domain over
V T , it is again a fundamental result of Cartan-Thullen-Oka theory [GuR] that there
exists a Stein Riemann domain
FT : U T → V T
which contains UT and such that O(U T )=O(UT ). ThemapPT : UT → T extends
to
PT : UT → T,
and UT can be identified with the open subset of U T composed by the connected
components of fibers of PT which cut Ω ⊂ U T . But, in fact, much better is true:
Lemma 2.6. Every fiber of PT is connected, that is:
UT = UT .
Proof. If not, then, by a connectivity argument, we may find a0,b0 ∈ P −1
T (t0),
ak,bk ∈ P −1
T (tk), with ak → a0 and bk → b0, such that:
(i) a0 ∈ �Lt0 , b0 ∈ P −1
T (t0) \ �Lt0 ;
(ii) ak,bk ∈ �Lt k .
Denote by Mt0 the maximal ideal of Ot0 (on T ), and for every p ∈ P−1 T (t0) denote by
Ip ⊂ Op the ideal generated by (PT ) ∗ (Mt0 ). At points of �Lt0 , this is just the ideal of
functions vanishing along �Lt0 ; whereas at points of P−1 T (t0) \ �Lt0 ,atwhichPTmay fail to be a submersion, this ideal may correspond to a “higher order” vanishing.
Because UT is Stein and P −1
T (t0) is a closed subvariety, we may find a holomorphic
function f ∈ O(U T ) such that:
(iii) f ≡ 0on�Lt0 , f ≡ 1onP−1 T (t0) \ �Lt0 ;
(iv) for every p ∈ P −1
T (t0), the differential dfp of f at p belongs to the ideal IpΩ 1 p.
Let {z1,...,zn−1} denote the coordinates on T lifted to UT . Then, by property (iv),
we can factorize
n−1
df =
∑
j=1
(z j − z j(t0)) · β j
where β j are holomorphic 1-forms on U T .
As in Lemma 2.5, each β j can be integrated along the simply connected fibers of
UT (with starting point on T), giving a function g j ∈ O(UT ). This function can be
holomorphically extended to the envelope UT . By the factorization above, and (ii),
we have
n−1
f (bk) − f (ak)=
∑
j=1
(z j(tk) − z j(t0)) · (g j(bk) − g j(ak))