NORMAL FAMILIES OF MEROMORPHIC MAPPINGS OF SEVERAL COMPLEX VARIABLES FOR MOVING HYPERSURFACES IN A COMPLEX PROJECTIVE SPACE

The main aim of this article is to give sufficient conditions for a family of meromorphic mappings of a domain D in C n into P N (C) to be meromorphically normal if they satisfy only some very weak conditions with respect to moving hypersurfaces in P N (C), namely that their intersections with these moving hypersurfaces, which may moreover depend on the meromorphic maps, are in some sense uniform. Our results generalize and complete previous results in this area, especially the works of Fujimoto 2, Tu 19, 20, TuLi 21, MaiThaiTrang 6 and the recent work of QuangTan 10.

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS

OF SEVERAL COMPLEX VARIABLES FOR MOVINGHYPERSURFACES IN A COMPLEX PROJECTIVE

SPACE

GERD DETHLOFF AND DO DUC THAI AND PHAM NGUYEN THU TRANG

Abstract The main aim of this article is to give sufficient

con-ditions for a family of meromorphic mappings of a domain D in

Cn into P N

(C) to be meromorphically normal if they satisfy only some very weak conditions with respect to moving hypersurfaces in

PN(C), namely that their intersections with these moving

hyper-surfaces, which may moreover depend on the meromorphic maps,

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

of Quang-Tan [10].

1 Introduction

Classically, a family F of holomorphic functions on a domain D ⊂ C

is said to be (holomorphically) normal if every sequence in F contains

a subsequence which converges uniformly on every compact subset of

D to a holomorphic map from D into P1

In 1957 Lehto and Virtanen [5] introduced the concept of normalmeromorphic functions in connection with the study of boundary be-haviour of meromorphic functions of one complex variable Since thennormal families of holomorphic maps have been studied intensively, re-sulting in an extensive development in the one complex variable contextand 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 pings of several complex variables were introduced by Rutishauser [11]and Stoll [14]

map-The research of the authors is partially supported by a NAFOSTED grant of Vietnam (Grant No 101.01.38.09).

1

The notion of a meromorphically normal family into the N sional complex projective space was introduced by H Fujimoto [2] (seesubsection 2.5 below for the definition of these concepts) Also in [2], hegave some sufficient conditions for a family of meromorphic mappings

-dimen-of a domain D in Cn into PN(C) to be meromorphically normal In

2002, Z Tu [20] considered meromorphically normal families of 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

mero-a fmero-amily of meromorphic mmero-appings of mero-a dommero-ain D in Cninto PN(C) forfixed hypersurfaces (see section 2 below for the necessary definitions):Theorem A ([6, Theorem A]) Let F be a family of meromorphicmappings of a domain D in Cn into PN(C) Suppose that for each

f ∈ F , there exist q ≥ 2N + 1 hypersurfaces H1(f ), H2(f ), , Hq(f ) in

PN(C) with

infD(H1(f ), , Hq(f )); f ∈ F > 0 and f (D) 6⊂ Hi(f ) (1 ≤ i ≤ N +1),where q is independent of f , but the hypersurfaces Hi(f ) may depend

on f , such that the following two conditions are satisfied:

i) For any fixed compact subset K of D, the 2(n − 1)-dimensionalLebesgue areas of f−1(Hi(f )) ∩ K (1 ≤ i ≤ N + 1) with countingmultiplicities are bounded above for all f in F

ii) There exists a closed subset S of D with Λ2n−1(S) = 0 such thatfor any fixed compact subset K of D − S, the 2(n − 1)-dimensionalLebesgue areas of f−1(Hi(f )) ∩ K (N + 2 ≤ i ≤ q) with countingmultiplicities are bounded above for all f in F

Then F is a meromorphically normal family on D

Recently, motivated by the investigation of Value Distribution 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

The-of meromorphic mappings The-of a domain D in Cn into PN(C) for movinghyperplanes or hypersurfaces has started While a substantial amount

of information has been amassed concerning the normality of families ofmeromorphic mappings for fixed targets through the years, the presentknowledge of this problem for moving targets has remained extremelymeagre 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 generalizesTheorem 2.2 of Tu-Li [21]:

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 3

Theorem B (see [10, Theorem 1.4]) Let F be a family of meromorphicmappings of a domain D ⊂ Cn into PN(C), and let Q1, · · · , Qq (q ≥2N +1) be q moving hypersurfaces in PN(C) in (weakly) general positionsuch that

i) For any fixed compact subset K of D, the 2(n − 1)-dimensionalLebesgue areas of f−1(Qj) ∩ K (1 ≤ j ≤ N + 1) counting multiplicitiesare uniformly bounded above for all f in F

ii) There exists a thin analytic subset S of D such that for any fixedcompact subset K of D, the 2(n − 1)-dimensional Lebesgue areas of

f−1(Qj) ∩ (K − S) (N + 2 ≤ j ≤ q) regardless of multiplicities areuniformly bounded above for all f in F

Then F is a meromorphically normal family on D

We would like to emphasize that, in Theorem B, the q moving persurfaces Q1, · · · , Qq in PN(C) are independent on f ∈ F (i.e theyare common for all f ∈ F ) Thus, the following question arised na-turally at this point: Does Theorem A hold for moving hypersurfaces

hy-H1(f ), H2(f ), , Hq(f ) which may depend on f ∈ F ? The main aim ofthis article is to give an affirmative answer to this question Namely,

we prove the following result which generalizes both Theorem A andTheorem B:

Theorem 1.1 Let F be a family of meromorphic mappings of a main D in Cn into PN(C) Suppose that for each f ∈ F, there exist

do-q ≥ 2N + 1 moving hypersurfaces H1(f ), H2(f ), , Hq(f ) in PN(C)such that the following three conditions are satisfied:

i) For each 1 6 k 6 q, the coefficients of the homogeneous mials Qk(f ) which define the Hk(f ) are bounded above uniformly oncompact subsets of D for all f in F , and for any sequence {f(p)} ⊂ F ,there exists z ∈ D (which may depend on the sequence) such that

polyno-infp∈ND(Q1(f(p)), , Qq(f(p)))(z) > 0 ii) For any fixed compact subset K of D, the 2(n − 1)-dimensionalLebesgue areas of f−1(Hi(f ))∩K (1 ≤ i ≤ N +1) counting multiplicitiesare bounded above for all f in F (in particular f (D) 6⊂ Hi(f ) (1 ≤ i ≤

N + 1))

iii) There exists a closed subset S of D with Λ2n−1(S) = 0 suchthat for any fixed compact subset K of D − S, the 2(n − 1)-dimensionalLebesgue areas of f−1(Hi(f ))∩K (N +2 ≤ i ≤ q) ignoring multiplicitiesare bounded above for all f in F

Then F is a meromorphically normal family on D.

In the special case of a family of holomorphic mappings, we get withthe same proof methods:

Theorem 1.2 Let F be a family of holomorphic mappings of a domain

D in Cn into PN(C) Suppose that for each f ∈ F, there exist q ≥2N + 1 moving hypersurfaces H1(f ), H2(f ), , Hq(f ) in PN(C) suchthat the following three conditions are satisfied:

i) For each 1 6 k 6 q, the coefficients of the homogeneous mials Qk(f ) which define the Hk(f ) are bounded above uniformly oncompact subsets of D for all f in F , and for any sequence {f(p)} ⊂ F ,there exists z ∈ D (which may depend on the sequence) such that

polyno-infp∈ND(Q1(f(p)), , Qq(f(p)))(z) > 0 ii) f (D) ∩ Hi(f ) = ∅ (1 6 i 6 N + 1) for any f ∈ F

iii) There exists a closed subset S of D with Λ2n−1(S) = 0 suchthat for any fixed compact subset K of D − S, the 2(n − 1)-dimensionalLebesgue areas of f−1(Hi(f ))∩K (N +2 ≤ i ≤ q) ignoring multiplicitiesare bounded above for all f in F

Then F is a holomorphically normal family on D

Remark 1.1 There are several examples in Tu [20] showing that theconditions 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 allownot to take into account at all the components of f−1(Hi(f )) of highorder:

The following theorem generalizes Theorem 2.1 of Tu-Li [21] fromthe case of moving hyperplanes which are independant of f to movinghypersurfaces which may depend on f (in fact observe that for mov-ing hyperplanes the condition H1, · · · , Hq in eS {Ti}N

i=0
is satisfied bytaking T0, , TN any (fixed or moving) N + 1 hyperplanes in generalposition)

Theorem 1.3 Let F be a family of holomorphic mappings of a main D in Cn into PN C
Let q > 2N + 1 be a positive integer Let

do-m1, · · · , mq be positive intergers or ∞ such that

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 5

Suppose that for each f ∈ F , there exist N + 1 moving hypersurfaces

i) For each 0 6 i 6 N, the coefficients of the homogeneous nomials Pi(f ) which define the Ti(f ) are bounded above uniformly oncompact subsets of D, and for all 1 6 j 6 q, the coefficients bij(f )

poly-of the linear combinations poly-of the Pi(f ), i = 0, , N which define thehomogeneous polynomials Qj(f ) which define the Hj(f ) are boundedabove uniformly on compact subsets of D, and for any fixed z ∈ D,

infD Q1 f
, · · · , Qq f

(z) : f ∈ F > 0

ii) f intersects Hj f
with multiplicity at least mj for each 1 ≤ j ≤ q(see subsection 2.6 for the necessary definitions)

Then F is a holomorphically normal family on D

The following theorem generalizes Theorem 1 of Tu [20] from thecase of fixed hyperplanes to moving hypersurfaces (in fact observe thatfor hyperplanes the condition H1(f ), · · · , Hq(f ) in eS {Ti(f )}N

i=0
is isfied by taking T0(f ), , TN(f ) any N + 1 hyperplanes in general po-sition)

sat-Theorem 1.4 Let F be a family of meromorphic mappings of a main D in Cn into PN C
Let q > 2N + 1 be a positive integer.Suppose that for each f ∈ F , there exist N + 1 moving hypersurfaces

i) For each 0 6 i 6 N, the coefficients of the homogeneous nomials Pi(f ) which define the Ti(f ) are bounded above uniformly oncompact subsets of D, and for all 1 6 j 6 q, the coefficients bij(f ) of thelinear combinations of the Pi(f ), i = 0, , N which define the homo-geneous polynomials Qj(f ) which define the Hj(f ) are bounded aboveuniformly on compact subsets of D, and for any sequence {f(p)} ⊂ F ,there exists z ∈ D (which may depend on the sequence) such that

poly-infp∈ND(Q1(f(p)), , Qq(f(p)))(z) > 0 ii) For any fixed compact K of D, the 2(n − 1)-dimensional Lebesgueareas of f−1 Hk(f )
∩ K (1 ≤ k ≤ N + 1) counting multiplicities are

bounded above for all f ∈ F (in particular f D
6⊂ Hk f
(1 ≤ k ≤

N + 1))

iii) There exists a closed subset S of D with Λ2n−1(S) = 0 such thatfor any fixed compact subset K of D − S, the 2(n − 1)-dimensionalLebesgue areas of

z ∈ Supp ν f, Hk(f )
ν f, Hk(f )
(z) < mk ∩ K (N + 2 ≤ k ≤ q)ignoring multiplicities for all f ∈ F are bounded above, where {mk}qk=N +2are fixed positive intergers or ∞ with

Then F is a meromorphically normal family on D

The following theorem generalizes Theorem 1 of Tu [19] from thecase of fixed hyperplanes to moving hypersurfaces

Theorem 1.5 Let F be a family of holomorphic mappings of a main D in Cn into PN C
Let q > 2N + 1 be a positive integer.Suppose that for each f ∈ F , there exist N + 1 moving hypersurfaces

i) For each 0 6 i 6 N, the coefficients of the homogeneous nomials Pi(f ) which define the Ti(f ) are bounded above uniformly oncompact subsets of D, and for all 1 6 j 6 q, the coefficients bij(f ) of thelinear combinations of the Pi(f ), i = 0, , N which define the homo-geneous polynomials Qj(f ) which define the Hj(f ) are bounded aboveuniformly on compact subsets of D, and for any sequence {f(p)} ⊂ F ,there exists z ∈ D (which may depend on the sequence) such that

poly-infp∈ND(Q1(f(p)), , Qq(f(p)))(z) > 0 ii) f (D) ∩ Hi(f ) = ∅ (1 6 i 6 N + 1) for any f ∈ F

iii) There exists a closed subset S of D with Λ2n−1(S) = 0 such thatfor any fixed compact subset K of D − S, the 2(n − 1)-dimensionalLebesgue areas of

{z ∈ Supp ν f, Hk(f )
ν f, Hk(f )
(z) < mk} ∩ K (N + 2 ≤ k ≤ q)

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 7

ignoring multiplicities for all f in F are bounded above, where {mk}qk=N +2are fixed positive intergers and may be ∞ with

Then F is a holomorphically normal family on D

Let us finally give some comments on our proof methods:

The proofs of Theorem 1.1 and Theorem 1.2 are obtained by eralizing ideas, which have been used by Thai-Mai-Trang [6] to proveTheorem A, to moving targets, which presents several highly non-trivialtechnical difficulties Among others, for a sequence of moving targetsH(f(p)) which at the same time may depend of the meromorphic maps

gen-f(p) : D → PN C
, obtaining a subsequence which converges locallyuniformly on D is much more difficult than for fixed targets (amongothers we cannot normalize the coefficients to have norm equal to 1everywhere like for fixed targets) This is obtained in Lemma 3.6, afterhaving proved in Lemma 3.5 that the condition D(Q1, , Qq) > δ > 0forces a uniform bound, only in terms of δ, on the degrees of the Qi,

1 ≤ i ≤ q (in fact the latter result fixes also a gap in [6] even for thecase of fixed targets)

The proofs of Theorem 1.3, Theorem 1.4 and Theorem 1.5 are tained by combining methods used by Tu [19], [20] and Tu-Li [21] withthe 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 thecase of hyperplanes, we still need that for every meromorphic map

ob-f(p) : D → PN C
, the Q1(f(p)), , Qq(f(p)) are still in a linear systemgiven by N + 1 such maps P0(f(p)), , PN(f(p)) The Lemmas 3.11 toLemma 3.14 adapt our technics to this situation (for example Lemma3.14 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 mapping Let U be a non-emptyconnected open subset of D A holomorphic mapping ˜f 6≡ 0 from Uinto CN +1 is said to be a representation of f on U if f (z) = ρ( ˜f (z))for all z ∈ U ∩ A − ˜f−1(0), where ρ : CN +1− {0} → PN(C) is thecanonical projection A holomorphic mapping f : A → PN(C) is said

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 Let f be a meromorphic mapping

of a domain D in Cn into PN(C) Then for any a ∈ D, f always has

an admissible representation ˜f (z) = (f0(z), f1(z), · · · , fN(z)) on someneighborhood U of a in D, which means that each fi(z) is a holomorphicfunction on U and f (z) = (f0(z) : f1(z) : · · · : fN(z)) outside theanalytic set I(f ) := {z ∈ U : f0(z) = f1(z) = = fN(z) = 0} ofcodimension ≥ 2

2.3 Moving hypersurfaces in general position Let D be a main in Cn Denote by HD the ring of all holomorphic functions

do-on D, and eHD[ω0, · · · , ωN] the set of all homogeneous polynomials

Q ∈ HD[ω0, · · · , ωN] such that the coefficients of Q are not all tically zero Each element of eHD[ω0, · · · , ωN] is said to be a movinghypersurface in PN(C)

iden-Let Q be a moving hypersurface of degree d > 1 Denote by Q(z)the homogeneous polynomial over CN +1 obtained by evaluating thecoefficients of Q in a specific point z ∈ D We remark that for generic

z ∈ D this is a non-zero homogenous polynomial with coefficients in C.The hypersurface H given by H(z) := {w ∈ CN +1: Q(z)(w) = 0} (forgeneric z ∈ D) is also called to be a moving hypersuface in PN(C) which

is defined by Q In this article, we identify Q with H if no confusionarises

We say that moving hypersurfaces {Qj}qj=1 of degree dj (q > N + 1)

in PN(C) are located in (weakly) general position if there exists z ∈ D

such that for any 1 6 j0 < · · · < jN 6 q, the system of equations

 Qji(z) ω0, · · · , ωN
= 0

0 6 i 6 Nhas only the trivial solution ω = 0, · · · , 0
in CN +1 This is equivalentto

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 9

2.4 Divisors Let D be a domain in Cn and f a non-identically zeroholomorphic function on D For a point a = (a1, a2, , an) ∈ D weexpand f as a compactly convergent series

homo-νf(a) := min{m; Pm(u) 6≡ 0}

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 thereare holomorphic functions g(z)(6≡ 0) and h(z)(6≡ 0) on a neighborhood

U of a with ν(z) = νg(z) − νh(z) on U We define the support of thedivisor ν on D by

Supp ν :={z ∈ D : ν(z) 6= 0}

We denote D+(D) = {ν : a non-negative divisor on D}

Let f be a meromorphic mapping from a domain D into PN C
.For each homogeneous polynomial Q ∈ eHD[ω0, · · · , ωN], we define thedivisor ν f, Q
on D as follows: For each a ∈ D, let f = fe 0, · · · , fN

be an admissible representation of f in a neighborhood U of a Then

we put

ν f, Q
(a) := νQ( ˜f )(a),where Q( ˜f ) := Q f0, · · · , fN

Let H be a moving hypersurface which is defined by the neous polynomial Q ∈ eHD[ω0, · · · , ωN], and f be a meromorphic map-ping of D into PN

homoge-C
As above we define the divisor ν(f, H)(z) :=

ν f, Q
(z) Obviously, Supp ν(f, H) is either an empty set or a pure(n − 1)−dimensional analytic set in D if f (D) 6⊂ H (i.e., Q( ˜f ) 6≡ 0 on

U ) We define ν(f, H) = ∞ on D and Supp ν(f, H) = D if f (D) ⊂ H.Sometimes we identify f−1(H) with the divisor ν(f, H) on D We canrewrite ν(f, H) as the formal sum ν(f, H) = P

i∈I

niXi, where Xi arethe irreducible components of Supp ν(f, H) and ni are the constantsν(f, H)(z) on Xi∩ Reg(Supp ν(f, H)), where Reg( ) denotes the set ofall the regular points

We say that the meromorphic mapping f intersects H with plicity at least m on D if ν(f, H)(z) ≥ m for all z ∈ Supp ν(f, H) and

multi-in particular that f multi-intersects H with multiplicity ∞ on D if f (D) ⊂ H

or f (D) ∩ H = ∅

2.5 Meromorphically normal families Let D be a domain in Cn.i) (See [1]) Let F be a family of holomorphic mappings of D into a com-pact complex manifold M F is said to be a (holomorphically) normalfamily on D if any sequence in F contains a subsequence which con-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 converge meromorphically on D to a meromorphicmapping f if and only if, for any z ∈ D, each f(p) has an admissiblerepresentation

˜(p) = (f0(p) : f1(p) : : fN(p))

on some fixed neighborhood U of z such that {fi(p)}∞

p=1 converges formly on compact subsets of U to a holomorphic function fi (0 ≤

uni-i ≤ N ) on U wuni-ith the property that ˜f = (f0 : f1 : : fN) is arepresentation of f on U (not necessarily an admissible one ! )

iii) (See [2]) Let F be a family of meromorphic mappings of D into

PN(C) F is said to be a meromorphically normal family on D if anysequence in F has a meromorphically convergent subsequence on D.iv) (See [14]) Let {νi} be a sequence of non-negative divisors on D

It is said to converge to a non-negative divisor ν on D if and only ifany a ∈ D has a neighborhood U such that there exist holomorphicfunctions hi(6≡ 0) and h(6≡ 0) on U such that νi = νhi, ν = νh and {hi}converges compactly to h on U

2.6 Other notations Let P0, · · · , PN be N + 1 homogeneous nomials of common degree in C[ω0, · · · , ωN] Denote by S {Pi}N

poly-i=0
theset of all homogeneous polynomials Q =

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 11

Let T0, · · · , TN be hypersurfaces in PN

C
of common degree, where

Ti is defined by the (not zero) polynomial Pi (0 6 i 6 N) Denote bye

S {Ti}N

i=0
the set of all hypersurfaces in PN

C
which are defined by

Q ∈ S {Pi}N

i=0
with Q not zero

Let P0, · · · , PN be N + 1 homogeneous polynomials of common gree in eHD[ω0, · · · , ωN] Denote by eS {Pi}N

de-i=0
the set of all neous not identically zero polynomials Q =

i=0
the set of all moving hypersurfaces in

PN C
which are defined by Q ∈ S {Pe i}N

i=0

Next, let Λd(S) denote the real d-dimensional Hausdorff measure of

S ⊂ Cn For a formal Z-linear combination X =P

i∈IniXi of analyticsubsets Xi ⊂ Cn and for a subset E ⊂ Cn, we call P

i∈IΛd(Xi∩ E)(resp P

i∈IniΛd(Xi∩ E)), the d-dimensional Lebesgue area of X ∩ Eignoring multiplicities (resp with counting multiplicities)

For each x ∈ Cn and R > 0, we set B(x, R) = {z ∈ Cn : ||z − x|| <R} and B(R) = B(0, R)

Denote by Hol(X, Y ) the set of all holomorphic mappings from acomplex space X to a complex space Y

3 Lemmas

Lemma 3.1 ([14, Theorem 2.24]) A sequence {νi} of non-negativedivisors on a domain D in Cnis normal in the sense of the convergence

of divisors on D if and only if the 2(n − 1)-dimensional Lebesgue areas

of νi∩ E (i ≥ 1) with counting multiplicities are bounded above for anyfixed compact set E of D

Lemma 3.2 ([14, Theorem 4.10]) If a sequence {νi} converges to ν in

D+(B(R)), then {supp νi} converges to supp ν (in the sense of closedsubsets of D, that means supp ν coincides with the set of all z such thatevery neighborhood U of z intersects supp νi for all but finitely many iand, simultaneously, with the set of all z such that every U intersectssupp νi for infinitely many i)

Lemma 3.3 ([14, Proposition 4.12]) Let {Ni} be a sequence of pure(n − 1)-dimensional analytic subsets of a domain D in Cn If the

2(n − 1)-dimensional Lebesgue areas of Ni∩ K ignoring multiplicities(i = 1, 2, ) are bounded above for any fixed compact subset K of D,then {Ni} is normal in the sense of the convergence of closed subsets

in D

Lemma 3.4 ([14, Proposition 4.11]) Let {Ni} be a sequence of pure(n−1)-dimensional analytic subsets of a domain D in Cn Assume thatthe 2(n−1)-dimensional Lebesgue areas of Ni∩K ignoring multiplicities(i = 1, 2, · · · ) are bounded above for any fixed compact subset K of Dand {Ni} converges to N as a sequence of closed subsets of D Then

N is either empty or a pure (n − 1)-dimensional analytic subset of D.Lemma 3.5 Let natural numbers N and q > N + 1 be fixed Then foreach δ > 0, there exists M (δ) = M (δ, N, q) > 0 such that the following

is satisfied:

For any homogeneous polynomials Q1, · · · , Qq on CN +1with complexcoefficients with norms bounded above by 1 such that D Q1, · · · , Qq
>

δ, we have max{deg Q1, · · · , deg Qq} < M (δ)

Proof First of all, we make the three following remarks

i) Let Q(ω) be a homogeneous polynomial on CN +1 such that

Q(ω) = X

|α|=d

aαωα,where |aα| ≤ 1 Then

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 13

iii) Since lim

infnD(Q(j)0 , , Q(j)N ) : j ≥ 1o> δ > 0,lim

j→∞

max

ndeg Q(j)0 , · · · , deg Q(j)N

o

= ∞

Without loss of generality we may assume that

deg Q(j)i = di ∀ 0 ≤ i ≤ k, ∀ j ≥ 1, anddeg Q(j)i = d(j)i → ∞ as j → ∞ for each k + 1 ≤ i ≤ N,

where k is some integer such that 0 ≤ k ≤ N − 1

Since deg Q(j)i = di ∀ 0 ≤ i ≤ k, ∀j ≥ 1, we may assume that, foreach 0 ≤ i ≤ k, nQ(j)i o

j≥1 converges uniformly on compact subsets

of CN +1 to either a homogeneous polynomial Qi of degree dj withcofficients being bounded above by 1 or to the zero polynomial Since

+) If k + 1 ≤ i ≤ N, then, by remark i), we have

Subcase 1.2 Assume that maxn|ω(0)0 |, , |ω(0)N |o= 1

We may assume that ω(0) = (1, 0, · · · , 0) Set ω(j)

This is a contradiction by the same argument as above

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 15

By Case 1, we have

max {deg Qj0, · · · , deg QjN} < M (δ/C, N, N + 1)

for any set {j0, , jN} ⊂ {1, , q} So if we define

M (δ, N, q) := M (δ/C, N, N + 1)(this is well defined since C only depends on N and q), then we have

max {deg Q1, · · · , deg Qq} < M (δ, N, q)

Lemma 3.6 Let natural numbers N and q > N + 1 be fixed Let

Hk(p) (1 6 k 6 q, p > 1) be moving hypersurfaces in PN

C
such thatthe following conditions are satisfied:

i) For each 1 6 k 6 q, p > 1, the coefficients of the homogeneouspolynomials Q(p)k which define the Hk(p) are bounded above uniformly oncompact subsets of D,

ii) there exists z0 ∈ D such that

infp∈ND(Q(p)

1 , , Q(p)q )(z0) > δ > 0 Then, we have:

a) There exists a subsequence {jp} ⊂ N such that for 1 6 k 6 q,

Q(jp )

k converge uniformly on compact subsets of D to not identicallyzero homogenous polynomials Qk (meaning that the Q(jp )

k and Qk arehomogenous polynomials in eHD[ω0, · · · , ωN] of the same degree, andall their coefficients converge uniformly on compact subsets of D).Moreover we have that D Q1, · · · , Qq
(z0) > δ > 0, the hypersurfaces

Q1(z0), · · · , Qq(z0) are located in general position and the moving persurfaces Q1(z), · · · , Qq(z) are located in (weakly) general position.b) There exists a subsequence {jp} ⊂ N and r = r(δ) > 0 such thatinf{D Q(jp )

hy-1 , · · · , Q(jp )

q
(z)

p > 1} > δ

4, ∀z ∈ B(z0, r).

Proof Let d(p)k = deg Q(p)k be the degree of the non identically vanishinghomogenous polynomial Q(p)k (1 6 k 6 q, p > 1) Then we have

holomor-k are boundedabove uniformly on compact subsets of D, there exists c > 0 such that

|akpI(z0)| ≤ c for all k, p, I Define homogenous polynomials

Since by equation (3.1) none of the homogenous polynomials Q(p)k (z0) (1 6

k 6 q, p > 1) can be the zero polynomial, we get that

max{deg ˜Q(p)1 (z), · · · , deg ˜Q(p)q (z)} < M (˜δ)for all z ∈ D So if again

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 17

can assume that {akpI(z)}∞p=1 converges uniformly on compact subsets

of D to akI for each k, I Denote by

Qk(z)(ω) = X

|I|=dk

akI(z).ωI.Then

D Q1, · · · , Qq
(z0) > lim inf

p−→∞ D Q(p)1 , · · · , Q(p)q
(z0) > δ > 0, (3.2)hence, the hypersurfaces Q1(z0), · · · , Qq(z0) are located in general po-sition and so the moving hypersurfaces Q1(z), · · · , Qq(z) are located in(weakly) general position (and in particular all the Q1(z), , Qq(z) arenot identically zero), which proves a)

Moreover, by equation (3.2), there exists r = r(δ) such that

Lemma 3.7 Let {f(p)} be a sequence of meromorphic mappings of

a domain D in Cn into PN(C) and let S be a closed subset of D with

Λ2n−1(S) = 0 Suppose that {f(p)} meromorphically converges on D−S

to a meromorphic mapping f of D − S into PN(C) Suppose that, foreach f(p), there exist N +1 moving hypersurfaces H1(f(p)), · · · , HN +1(f(p))

in PN(C), where the moving hypersurfaces Hi(f(p)) may depend on f(p),such that the following three conditions are satisfied:

i) For each 1 6 k 6 N + 1, the coefficients of homogeneous nomial Qk(f(p)) which define Hk(f(p)) for all f(p) are bounded aboveuniformly on compact subsets of D

poly-ii) There exists z0 ∈ D such that

NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 7

ignoring multiplicities for all f in F are... Meromorphically normal families Let D be a domain in Cn.i) (See [1]) Let F be a family of holomorphic mappings of D into a com-pact complex manifold M F is said to be a (holomorphically) normalfamily... subsets of a domain D in Cn If the

2(n − 1)-dimensional Lebesgue areas of Ni∩