Table Of ContentNORMAL FAMILIES OF MEROMORPHIC MAPPINGS
OF SEVERAL COMPLEX VARIABLES FOR MOVING
HYPERSURFACES IN A COMPLEX PROJECTIVE
3 SPACE
1
0
2 GERDDETHLOFFAND DO DUC THAI AND PHAMNGUYENTHU TRANG
n
a
J
4
Abstract. The main aim of this article is to give sufficient con-
V] ditions for a family of meromorphic mappings of a domain D in
Cn into PN(C) to be meromorphically normal if they satisfy only
C
someveryweakconditionswithrespecttomovinghypersurfacesin
.
h PN(C), namely that their intersections with these moving hyper-
t
a surfaces, which may moreover depend on the meromorphic maps,
m are in some sense uniform. Our results generalize and complete
[ previous results in this area, especially the works of Fujimoto [2],
Tu [19], [20], Tu-Li [21], Mai-Thai-Trang [6] and the recent work
1
of Quang-Tan [10].
v
7
8
6 1. Introduction.
0
.
1
0 Classically, a family of holomorphic functions on a domain D C
F ⊂
3 is said to be (holomorphically) normal if every sequence in contains
1 F
a subsequence which converges uniformly on every compact subset of
:
v D to a holomorphic map from D into P1.
i
X
In 1957 Lehto and Virtanen [5] introduced the concept of normal
r
a meromorphic functions in connection with the study of boundary be-
haviour of meromorphic functions of one complex variable. Since then
normal families of holomorphic maps have been studied intensively, re-
sultinginanextensive development intheonecomplexvariablecontext
and in generalizations to the several complex variables setting (see [22],
[3], [4], [1] and the references cited in [22] and [4]).
The first ideas and results on normal families of meromorphic map-
pings of several complex variables were introduced by Rutishauser [11]
and Stoll [14].
The research of the authors is partially supported by a NAFOSTED grant of
Vietnam (Grant No. 101.01.38.09).
1
2 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG
The notion of a meromorphically normal family into the N-dimen-
sional complex projective space was introduced by H. Fujimoto [2] (see
subsection 2.5belowforthedefinitionoftheseconcepts). Also in[2], he
gave some sufficient conditions for a family of meromorphic mappings
of a domain D in Cn into PN(C) to be meromorphically normal. In
2002, Z. Tu [20] considered meromorphically normal families of mero-
morphic mappings of a domain D in Cn into PN(C) for hyperplanes.
Generalizing the above results of Fujimoto and Tu, in 2005, Thai-Mai-
Trang [6] gave a sufficient condition for the meromorphic normality of
a family of meromorphic mappings of a domain D in Cn into PN(C) for
fixed hypersurfaces (see section 2 below for the necessary definitions):
Theorem A. ([6, Theorem A]) Let be a family of meromorphic
mappings of a domain D in Cn intoFPN(C). Suppose that for each
f , there exist q 2N +1 hypersurfaces H (f),H (f),...,H (f) in
1 2 q
PN∈(CF) with ≥
inf D(H (f),...,H (f));f > 0 and f(D) H (f)(1 i N+1),
1 q i
∈ F 6⊂ ≤ ≤
(cid:8) (cid:9)
where q is independent of f, but the hypersurfaces H (f) may depend
i
on f, such that the following two conditions are satisfied:
i) For any fixed compact subset K of D, the 2(n 1)-dimensional
−
Lebesgue areas of f 1(H (f)) K (1 i N + 1) with counting
− i
∩ ≤ ≤
multiplicities are bounded above for all f in .
F
ii) There exists a closed subset S of D with Λ2n 1(S) = 0 such that
−
for any fixed compact subset K of D S, the 2(n 1)-dimensional
− −
Lebesgue areas of f 1(H (f)) K (N + 2 i q) with counting
− i
∩ ≤ ≤
multiplicities are bounded above for all f in .
F
Then is a meromorphically normal family on D.
F
Recently, motivated by the investigation of Value Distribution The-
ory for moving hyperplanes (for example Ru and Stoll [12], [13], Stoll
[15], and Thai-Quang [16], [17]), the study of the normality of families
of meromorphic mappings of a domain D in Cn into PN(C) for moving
hyperplanes or hypersurfaces has started. While a substantial amount
ofinformationhasbeenamassedconcerning thenormalityoffamiliesof
meromorphic mappings for fixed targets through the years, the present
knowledge of this problem for moving targets has remained extremely
meagre. There are only a few such results in some restricted situations
(see [21], [10]). For instance, we recall a recent result of Quang-Tan
[10] which is the best result available at present and which generalizes
Theorem 2.2 of Tu-Li [21]:
NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 3
Theorem B.(see [10, Theorem1.4])Let be a family of meromorphic
mappings of a domain D Cn into PN(FC), and let Q , ,Q (q
1 q
2N+1) be q moving hypersu⊂rfaces in PN(C) in (weakly) gen·e·r·al positio≥n
such that
i) For any fixed compact subset K of D, the 2(n 1)-dimensional
−
Lebesgue areas of f 1(Q ) K (1 j N +1) counting multiplicities
− j
∩ ≤ ≤
are uniformly bounded above for all f in .
F
ii) There exists a thin analytic subset S of D such that for any fixed
compact subset K of D, the 2(n 1)-dimensional Lebesgue areas of
−
f 1(Q ) (K S) (N + 2 j q) regardless of multiplicities are
− j
∩ − ≤ ≤
uniformly bounded above for all f in .
F
Then is a meromorphically normal family on D.
F
We would like to emphasize that, in Theorem B, the q moving hy-
persurfaces Q , ,Q in PN(C) are independent on f (i.e. they
1 q
··· ∈ F
are common for all f .) Thus, the following question arised na-
∈ F
turally at this point: Does Theorem A hold for moving hypersurfaces
H (f),H (f),...,H (f) which may depend on f ? The main aim of
1 2 q
∈ F
this article is to give an affirmative answer to this question. Namely,
we prove the following result which generalizes both Theorem A and
Theorem B:
Theorem 1.1. Let be a family of meromorphic mappings of a do-
main D in Cn into PFN(C). Suppose that for each f , there exist
q 2N + 1 moving hypersurfaces H (f),H (f),...,H∈ F(f) in PN(C)
1 2 q
≥
such that the following three conditions are satisfied:
i) For each 1 6 k 6 q, the coefficients of the homogeneous polyno-
mials Q (f) which define the H (f) are bounded above uniformly on
k k
compact subsets of D for all f in , and for any sequence f(p) ,
F { } ⊂ F
there exists z D (which may depend on the sequence) such that
∈
infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0.
∈
(cid:8) (cid:9)
ii) For any fixed compact subset K of D, the 2(n 1)-dimensional
−
Lebesgueareas of f 1(H (f)) K (1 i N+1) counting multiplicities
− i
∩ ≤ ≤
are bounded above for all f in (in particular f(D) H (f) (1 i
i
F 6⊂ ≤ ≤
N +1)).
iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such
−
that for any fixed compact subset K of D S, the 2(n 1)-dimensional
− −
Lebesgue areas of f 1(H (f)) K (N+2 i q) ignoring multiplicities
− i
∩ ≤ ≤
are bounded above for all f in .
F
4 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG
Then is a meromorphically normal family on D.
F
In the special case of a family of holomorphic mappings, we get with
the same proof methods:
Theorem 1.2. Let be a family of holomorphicmappings of a domain
D in Cn into PN(CF). Suppose that for each f , there exist q
2N + 1 moving hypersurfaces H (f),H (f),...,H∈ (Ff) in PN(C) suc≥h
1 2 q
that the following three conditions are satisfied:
i) For each 1 6 k 6 q, the coefficients of the homogeneous polyno-
mials Q (f) which define the H (f) are bounded above uniformly on
k k
compact subsets of D for all f in , and for any sequence f(p) ,
F { } ⊂ F
there exists z D (which may depend on the sequence) such that
∈
infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0.
∈
(cid:8) (cid:9)
ii) f(D) H (f) = (1 6 i 6 N +1) for any f F .
i
∩ ∅ ∈
iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such
−
that for any fixed compact subset K of D S, the 2(n 1)-dimensional
− −
Lebesgue areas of f 1(H (f)) K (N+2 i q) ignoring multiplicities
− i
∩ ≤ ≤
are bounded above for all f in .
F
Then is a holomorphically normal family on D.
F
Remark 1.1. There are several examples in Tu [20] showing that the
conditions in i), ii) and iii) in Theorem 1.1 and Theorem 1.2 cannot
be omitted.
We also generalise several results of Tu [19], [20], [21] which allow
not to take into account at all the components of f 1(H (f)) of high
− i
order:
The following theorem generalizes Theorem 2.1 of Tu-Li [21] from
the case of moving hyperplanes which are independant of f to moving
hypersurfaces which may depend on f (in fact observe that for mov-
ing hyperplanes the condition H , ,H in T N is satisfied by
1 ··· q S { i}i=0
taking T ,...,T any (fixed or moving) N + 1(cid:0)hyperpl(cid:1)anes in general
0 N
e
position).
Theorem 1.3. Let be a family of holomorphic mappings of a do-
main D in Cn into PFN C . Let q > 2N +1 be a positive integer. Let
m , ,m be positive (cid:0)int(cid:1)ergers or such that
1 q
··· ∞
q
N
1 > N +1.
(cid:18) − m (cid:19)
Xj=1 j
NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 5
Suppose that for each f , there exist N + 1 moving hypersurfaces
T f , ,T f in PN∈CFof common degree and there exist q mov-
0 N
···
ing(cid:0) h(cid:1)ypersurfa(cid:0)ces(cid:1) H f (cid:0), (cid:1) ,H f in T f N such that the
1 ··· q S { i }i=0
following conditions a(cid:0)re(cid:1)satisfied:(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1)
e
i) For each 0 6 i 6 N, the coefficients of the homogeneous poly-
nomials P (f) which define the T (f) are bounded above uniformly on
i i
compact subsets of D, and for all 1 6 j 6 q, the coefficients b (f)
ij
of the linear combinations of the P (f), i = 0,...,N which define the
i
homogeneous polynomials Q (f) which define the H (f) are bounded
j j
above uniformly on compact subsets of D, and for any fixed z D,
∈
inf D Q f , ,Q f (z) : f > 0.
1 q
··· ∈ F
(cid:8) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) (cid:9)
ii) f intersects H f with multiplicity at least m for each 1 j q
j j
≤ ≤
(see subsection 2.6 fo(cid:0)r (cid:1)the necessary definitions).
Then is a holomorphically normal family on D.
F
The following theorem generalizes Theorem 1 of Tu [20] from the
case of fixed hyperplanes to moving hypersurfaces (in fact observe that
forhyperplanestheconditionH (f), ,H (f)in T (f) N issat-
1 ··· q S { i }i=0
isfied by taking T (f),...,T (f) any N +1 hyperpla(cid:0)nes in gene(cid:1)ral po-
0 N
e
sition).
Theorem 1.4. Let be a family of meromorphic mappings of a do-
main D in Cn intoFPN C . Let q > 2N + 1 be a positive integer.
Suppose that for each f (cid:0) (cid:1) , there exist N + 1 moving hypersurfaces
T f , ,T f in PN∈CFof common degree and there exist q mov-
0 N
···
ing(cid:0) h(cid:1)ypersurfa(cid:0)ces(cid:1) H f (cid:0), (cid:1) ,H f in T f N such that the
1 ··· q S { i }i=0
following conditions a(cid:0)re(cid:1)satisfied:(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1)
e
i) For each 0 6 i 6 N, the coefficients of the homogeneous poly-
nomials P (f) which define the T (f) are bounded above uniformly on
i i
compactsubsets of D, and for all 1 6 j 6 q, the coefficients b (f) of the
ij
linear combinations of the P (f), i = 0,...,N which define the homo-
i
geneous polynomials Q (f) which define the H (f) are bounded above
j j
uniformly on compact subsets of D, and for any sequence f(p) ,
{ } ⊂ F
there exists z D (which may depend on the sequence) such that
∈
infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0.
∈
(cid:8) (cid:9)
ii) For any fixed compact K of D, the 2(n 1)-dimensional Lebesgue
−
areas of f 1 H (f) K (1 k N +1) counting multiplicities are
− k
∩ ≤ ≤
(cid:0) (cid:1)
6 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG
bounded above for all f (in particular f D H f (1 k
k
∈ F 6⊂ ≤ ≤
N +1)). (cid:0) (cid:1) (cid:0) (cid:1)
iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such that
−
for any fixed compact subset K of D S, the 2(n 1)-dimensional
− −
Lebesgue areas of
z Suppν f,H (f) ν f,H (f) (z) < m K (N +2 k q)
k k k
∈ ∩ ≤ ≤
(cid:8) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1) (cid:9)
ignoringmultiplicities fo(cid:12)rallf are boundedabove, where m q
∈ F { k}k=N+2
are fixed positive intergers or with
∞
q
1 q N +1
< − .
m (cid:0)N (cid:1)
X k
k=N+2
Then is a meromorphically normal family on D.
F
The following theorem generalizes Theorem 1 of Tu [19] from the
case of fixed hyperplanes to moving hypersurfaces.
Theorem 1.5. Let be a family of holomorphic mappings of a do-
main D in Cn intoFPN C . Let q > 2N + 1 be a positive integer.
Suppose that for each f (cid:0) (cid:1) , there exist N + 1 moving hypersurfaces
T f , ,T f in PN∈CFof common degree and there exist q mov-
0 N
···
ing(cid:0) h(cid:1)ypersurfa(cid:0)ces(cid:1) H f (cid:0), (cid:1) ,H f in T f N such that the
1 ··· q S { i }i=0
following conditions a(cid:0)re(cid:1)satisfied:(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1)
e
i) For each 0 6 i 6 N, the coefficients of the homogeneous poly-
nomials P (f) which define the T (f) are bounded above uniformly on
i i
compactsubsets of D, and for all 1 6 j 6 q, the coefficients b (f) of the
ij
linear combinations of the P (f), i = 0,...,N which define the homo-
i
geneous polynomials Q (f) which define the H (f) are bounded above
j j
uniformly on compact subsets of D, and for any sequence f(p) ,
{ } ⊂ F
there exists z D (which may depend on the sequence) such that
∈
infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0.
∈
(cid:8) (cid:9)
ii) f(D) H (f) = (1 6 i 6 N +1) for any f .
i
∩ ∅ ∈ F
iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such that
−
for any fixed compact subset K of D S, the 2(n 1)-dimensional
− −
Lebesgue areas of
z Suppν f,H (f) ν f,H (f) (z) < m K (N +2 k q)
k k k
{ ∈ }∩ ≤ ≤
(cid:0) (cid:1)(cid:12) (cid:0) (cid:1)
(cid:12)
NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 7
ignoringmultiplicities forallf in areboundedabove, where m q
F { k}k=N+2
are fixed positive intergers and may be with
∞
q
1 q N +1
< − .
m (cid:0)N (cid:1)
X k
k=N+2
Then is a holomorphically normal family on D.
F
Let us finally give some comments on our proof methods:
The proofs of Theorem 1.1 and Theorem 1.2 are obtained by gen-
eralizing ideas, which have been used by Thai-Mai-Trang [6] to prove
TheoremA,tomovingtargets, whichpresentsseveralhighlynon-trivial
technical difficulties. Among others, for a sequence of moving targets
H(f(p)) which at the same time may depend of the meromorphic maps
f(p) : D PN C , obtaining a subsequence which converges locally
→
uniformly on D(cid:0)is(cid:1)much more difficult than for fixed targets (among
others we cannot normalize the coefficients to have norm equal to 1
everywhere like for fixed targets). This is obtained in Lemma 3.6, after
having proved in Lemma 3.5 that the condition D(Q ,...,Q ) > δ > 0
1 q
forces a uniform bound, only in terms of δ, on the degrees of the Q ,
i
1 i q (in fact the latter result fixes also a gap in [6] even for the
≤ ≤
case of fixed targets).
The proofs of Theorem 1.3, Theorem 1.4 and Theorem 1.5 are ob-
tained by combining methods used by Tu [19], [20] and Tu-Li [21] with
the methods which we developed to prove our first two theorems. How-
ever, in order to apply the technics which Tu and Tu-Li used for the
case of hyperplanes, we still need that for every meromorphic map
f(p) : D PN C , the Q (f(p)),...,Q (f(p)) are still in a linear system
1 q
→
given by N +(cid:0)1 s(cid:1)uch maps P (f(p)),...,P (f(p)). The Lemmas 3.11 to
0 N
Lemma 3.14 adapt our technics to this situation (for example Lemma
3.14 is an adaptation of our Lemma 3.6)
2. Basic notions.
2.1. Meromorphic mappings. Let A be a non-empty open subset
of a domain D in Cn such that S = D A is an analytic set in D.
Let f : A PN(C) be a holomorphic map−ping. Let U be a non-empty
→
˜
connected open subset of D. A holomorphic mapping f 0 from U
into CN+1 is said to be a representation of f on U if f(z6≡) = ρ(f˜(z))
for all z U A f˜ 1(0), where ρ : CN+1 0 PN(C) is the
−
canonical∈proje∩ction−. A holomorphic mapping−f {: A} →PN(C) is said
→
8 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG
to be a meromorphic mapping from D into PN(C) if for each z D,
∈
there exists a representation of f on some neighborhood of z in D.
2.2. Admissible representations. Letf beameromorphic mapping
of a domain D in Cn into PN(C). Then for any a D, f always has
∈
˜
an admissible representation f(z) = (f (z),f (z), ,f (z)) on some
0 1 N
···
neighborhoodU ofainD,whichmeansthateachf (z)isaholomorphic
i
function on U and f(z) = (f (z) : f (z) : : f (z)) outside the
0 1 N
···
analytic set I(f) := z U : f (z) = f (z) = ... = f (z) = 0 of
0 1 N
{ ∈ }
codimension 2.
≥
2.3. Moving hypersurfaces in general position. Let D be a do-
main in Cn. Denote by the ring of all holomorphic functions
D
H
on D, and [ω , ,ω ] the set of all homogeneous polynomials
D 0 N
H ···
Q [ω , ,ω ] such that the coefficients of Q are not all iden-
D 0 N
∈ H ·e··
tically zero. Each element of [ω , ,ω ] is said to be a moving
D 0 N
hypersurface in PN(C). H ···
e
Let Q be a moving hypersurface of degree d > 1. Denote by Q(z)
the homogeneous polynomial over CN+1 obtained by evaluating the
coefficients of Q in a specific point z D. We remark that for generic
z D this is a non-zero homogenous p∈olynomial with coefficients in C.
Th∈e hypersurface H given by H(z) := w CN+1 : Q(z)(w) = 0 (for
genericz D)isalsocalledtobeamov{ing∈hypersufaceinPN(C)w}hich
∈
is defined by Q. In this article, we identify Q with H if no confusion
arises.
We say that moving hypersurfaces Q q of degree d (q > N+1)
{ j}j=1 j
in PN(C) are located in (weakly) general position if there exists z D
∈
such that for any 1 6 j < < j 6 q, the system of equations
0 N
···
Q (z) ω , ,ω = 0
ji 0 ··· N
(cid:26) 0 6 i 6(cid:0) N (cid:1)
has only the trivial solution ω = 0, ,0 in CN+1. This is equivalent
···
to (cid:0) (cid:1)
2 2
D(Q ,...,Q )(z) := inf Q (z)(ω) + + Q (z)(ω) > 0,
1 q 1≤j0<Y···<jN≤q||ω||=1(cid:18)(cid:12)(cid:12) j0 (cid:12)(cid:12) ··· (cid:12)(cid:12) jN (cid:12)(cid:12) (cid:19)
where Q (z)(ω) = a (z).ωI and ω = ω 2 1/2.
j I =dj jI || || | j|
| |
P (cid:0)P (cid:1)
NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 9
2.4. Divisors. Let D be a domain in Cn and f a non-identically zero
holomorphic function on D. For a point a = (a ,a ,...,a ) D we
1 2 n
∈
expand f as a compactly convergent series
∞
f(u +a ,.....,u +a ) = P (u ,...,u )
1 1 n n m 1 n
mX=0
on a neighborhood of a, where P is either identically zero or a homo-
m
geneous polynomial of degree m. The number
ν (a) := min m;P (u) 0
f m
{ 6≡ }
is said to be the zero multiplicity of f at a. By definition, a divisor on
D is an integer-valued function ν on D such that for every a D there
∈
are holomorphic functions g(z)( 0) and h(z)( 0) on a neighborhood
6≡ 6≡
U of a with ν(z) = ν (z) ν (z) on U. We define the support of the
g h
−
divisor ν on D by
Supp ν := z D : ν(z) = 0 .
{ ∈ 6 }
We denote +(D) = ν : a non-negative divisor on D .
D { }
Let f be a meromorphic mapping from a domain D into PN C .
For each homogeneous polynomial Q [ω , ,ω ], we define(cid:0)th(cid:1)e
D 0 N
∈ H ···
divisor ν f,Q on D as follows: For each a D, let f = f , ,f
e ∈ 0 ··· N
be an ad(cid:0)missib(cid:1)le representation of f in a neighborhood U(cid:0)of a. Then(cid:1)
e
we put
ν f,Q (a) := ν (a),
Q(f˜)
(cid:0) (cid:1)
˜
where Q(f) := Q f , ,f .
0 N
···
(cid:0) (cid:1)
Let H be a moving hypersurface which is defined by the homoge-
neous polynomial Q [ω , ,ω ], and f be a meromorphic map-
D 0 N
ping of D into PN C∈.HAs abo·v·e· we define the divisor ν(f,H)(z) :=
e
ν f,Q (z). Obviou(cid:0)sly(cid:1), Suppν(f,H) is either an empty set or a pure
(n(cid:0) 1)(cid:1) dimensional analytic set in D if f(D) H (i.e., Q(f˜) 0 on
− − 6⊂ 6≡
U). We define ν(f,H) = on D and Suppν(f,H) = D if f(D) H.
∞ ⊂
Sometimes we identify f 1(H) with the divisor ν(f,H) on D. We can
−
rewrite ν(f,H) as the formal sum ν(f,H) = n X , where X are
i i i
iP∈I
the irreducible components of Suppν(f,H) and n are the constants
i
ν(f,H)(z) on X Reg(Suppν(f,H)), where Reg( ) denotes the set of
i
∩
all the regular points.
We say that the meromorphic mapping f intersects H with multi-
plicity at least m on D if ν(f,H)(z) m for all z Suppν(f,H) and
≥ ∈
10GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG
inparticular that f intersects H withmultiplicity onD if f(D) H
∞ ⊂
or f(D) H = .
∩ ∅
2.5. Meromorphically normal families. Let D be a domain in Cn.
i)(See [1]) Let bea familyof holomorphicmappings ofD into acom-
F
pact complex manifold M. is said to be a (holomorphically) normal
F
family on D if any sequence in contains a subsequence which con-
F
verges uniformly on compact subsets of D to a holomorphic mapping
of D into M.
ii) (See [2]) A sequence f(p) of meromorphic mappings from D into
PN(C) is said to conver{ge m}eromorphically on D to a meromorphic
mapping f if and only if, for any z D, each f(p) has an admissible
∈
representation
f˜(p) = (f(p) : f(p) : ... : f(p))
0 1 N
(p)
on some fixed neighborhood U of z such that f converges uni-
{ i }∞p=1
formly on compact subsets of U to a holomorphic function f (0
i
≤
˜
i N) on U with the property that f = (f : f : ... : f ) is a
0 1 N
≤
representation of f on U (not necessarily an admissible one ! ).
iii) (See [2]) Let be a family of meromorphic mappings of D into
PN(C). is saidFto be a meromorphically normal family on D if any
F
sequence in has a meromorphically convergent subsequence on D.
F
iv) (See [14]) Let ν be a sequence of non-negative divisors on D.
i
{ }
It is said to converge to a non-negative divisor ν on D if and only if
any a D has a neighborhood U such that there exist holomorphic
∈
functions h ( 0) and h( 0) on U such that ν = ν , ν = ν and h
i 6≡ 6≡ i hi h { i}
converges compactly to h on U.
2.6. Other notations. Let P , ,P be N +1 homogeneous poly-
0 N
nomials of common degree in C[ω··,· ,ω ]. Denote by P N the
0 ··· N S { i}i=0
N (cid:0) (cid:1)
set of all homogeneous polynomials Q = b P (b C).
i i i
∈
i=0
P
N
Let Q := b P q be q (q > N +1) homogeneous polynomials
{ j ji i}j=1
i=0
P
in P N . We say that Q q are located in general position in
S { i}i=0 { j}j=1
P(cid:0) N if(cid:1)
S { i}i=0
(cid:0) (cid:1)
1 6 j < < j 6 q, det b = 0.
∀ 0 ··· N jki 06k,i6N 6
(cid:0) (cid:1)
