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Normal families of meromorphicmappings sharing hypersurfacesNguyen Thi Thu Hanga & Tran Van Tana

a Department of Mathematics, Hanoi National University ofEducation, 136- Xuan Thuy Street, Cau Giay, Hanoi, Vietnam.Published online: 01 May 2014.

To cite this article: Nguyen Thi Thu Hang & Tran Van Tan (2014): Normal families of meromorphicmappings sharing hypersurfaces, Complex Variables and Elliptic Equations: An International Journal,DOI: 10.1080/17476933.2014.911292

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Complex Variables and Elliptic Equations, 2014http://dx.doi.org/10.1080/17476933.2014.911292

Normal families of meromorphic mappings sharing hypersurfaces

Nguyen Thi Thu Hang and Tran Van Tan∗

Department of Mathematics, Hanoi National University of Education, 136- Xuan Thuy Street, CauGiay, Hanoi, Vietnam

Communicated by S. Ivashkovich

(Received 15 August 2013; accepted 30 March 2014)

We study the normality of families of meromorphic mappings of a domainD ⊂ C

m into the complex projective space CPn having the same inverse image(ignoring counting multiplicities) of moving hypersurfaces.

Keywords: normal family; meromorphic mappings; moving hypersurfaces;Nevanlinna theory

AMS Subject Classifications: 32A10; 32C10; 32H20

1. Introduction

The Little Picard Theorem states that if a meromorphic function on the complex plane C

omits three distinct points in C, then it is a constant function; and the classical result ofMontel says that the family F of meromorphic functions on a domain D ⊂ C is normalif there are three distinct points a, b, c ∈ C such that each element of F omits each ofa, b and c in D. The Little Picard Theorem was generalized to the case of entire curves inthe complement of 2n + 1 hyperplanes in general position in CPn by Green [1], and tothe case of entire curves in the complement of 2n + 1 hypersurfaces in general positionin CPn by Eremenko [2]. According to Bloch’s principle, to every ‘Picard-type’ theorem,there should belong a corresponding normality criterion. The normality result correspondingto the aforementioned Picard-type theorems was proved by Tu [3], and Tu-Li [4]. In thispaper, we examine this problem for the case where the mappings of the family can meet thehyperplanes (and hypersurfaces).

Let f be a meromorphic mapping of a domain D in Cm into CPn . Then for each a ∈ D,

f has a reduced representation f̃ = ( f0, · · · , fn) on a neighbourhood U of a in D whichmeans that each fi is a holomorphic function on U and f (z) = ( f0(z) : · · · : fn(z))outside the analytic set (of all points of indetermination of f ) I ( f ) := {z : f0(z) = · · · =fn(z) = 0} of codimension ≥ 2.

Denote by HD the ring of all holomorphic functions on D. Let Q be a homoge-neous polynomial in HD[x0, . . . , xn] of degree d ≥ 1. Denote by Q(z) the homogeneouspolynomial over C obtained by substituting a specific point z ∈ D into the coefficientsof Q. We define a moving hypersurface in CPn to be any homogeneous polynomial

∗Corresponding author. Email: thuhanghp2003@gmail.com

© 2014 Taylor & Francis

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2 N.T.T. Hang and T.V. Tan

Q ∈ HD[x0, . . . , xn] such that the coefficients of Q have no common zero point. We saythat moving hypersurfaces {Q j }q

j=1 (q ≥ t + 1) in CPn are in t−subgeneral position withrespect to a subset X ⊂ CPn, if there exists z ∈ D such that for any 1 ≤ j0 < · · · < jt ≤ qthe system of equations {

Q ji (z)(w0, . . . , wn) = 00 ≤ i ≤ t

has no solution (w0, . . . , wn) �= (0, . . . , 0) satisfying (w0 : · · · : wn) ∈ X.

Let Q be a moving hypersurface in CPn, and let f be a meromorphic mapping of D intoCPn . For z ∈ D, take f̃ = ( f0, . . . , fn) is a reduced representation of f in a neighbourhoodof U of z.The divisor ν( f,Q)(z) := νQ( f̃ )

(z)(z ∈ U ) is determined independently of a choiceof reduced representations, and hence is well defined on the totality of D. Here, νQ( f̃ )

is

the zero divisor of holomorphic function Q( f̃ ).

In 1974, Fujimoto [5] introduced the notion of a meromorphically normal family intothe complex projective space.

A sequence { fk}∞k=1 of meromorphic mappings of a domain D in Cm into CPn is said

to converge meromorphically on D to a meromorphic mapping f of D into CPn if andonly if, for any z ∈ D, each fk has a reduced representation f̃k = ( fk0, · · · , fkn) on somefixed neighbourhood U of z such that { fki }∞k=1 converges uniformly on compact subsets ofU to a holomorphic function fi (0 ≤ i ≤ n) on U with the property that ( f0, · · · , fn) is arepresentation of f in U.

A family F of meromorphic mappings of a domain D in Cm into CPn is said to be

meromorphically normal on D if any sequence in F has a meromorphically convergentsubsequence on D.

In [5], Fujimoto obtained the following interesting result.

Theorem F Let F be a family of meromorphic mappings of a domain D ⊂ Cm into CPn

and let {Hj }(2n+1)j=1 be hyperplanes in CPn in general position such that for each f ∈ F ,

f (D) �⊂ Hj ( j = 1, . . . , 2n + 1) and for any fixed compact subset K of D, the 2(m − 1)-dimensional Lebesgue areas of f −1(Hj )∩ K ( j = 1, . . . , 2n+1) inclusive of multiplicitiesfor all f in F are bounded above by a fixed constant. Then F is a meromorphically normalfamily on D.

We refer the readers to [3,4,6,7,9], for extensions of the above theorem for both casesof fixed and moving hypersurfaces. I would like to remark that in all of these results, themultiplicity of intersections is taken into account.

In this paper, we obtain the following results, where multiplicity of intersections isdisregarded.

Theorem 1.1 Let X ⊂ CPn be a projective variety. Let Q1, . . . , Q2t+1 be movinghypesurfaces in CPn in t-subgeneral position with respect to X. Let F be a family ofmeromorphic mappings f of a domain D ⊂ C

m into X, such that Q j ( f ) �≡ 0, for allj ∈ {1, . . . , 2t + 1}. Assume that

(a) f −1(Q j ) = g−1(Q j ) (as sets) for all f, g in F , and for all j ∈ {1, . . . , 2t + 1},(b) dim(∩2t+1

j=1 f −1(Qi )) ≤ m − 2 for f ∈ F .

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Complex Variables and Elliptic Equations 3

Then F is a meromorphically normal family on D.

For each moving hyperplane H in CPn defined by the homogeneous polynomial H :=a0x0 + · · · + an xn ∈ HD[x0, . . . , xn] (the coefficients a0, . . . , an have no common zeropoint), we define a holomorphic map H∗ of D into CPn with the reduced representation(a0 : · · · : an). Let 2t + 1(t ≥ n) moving hyperplanes Hj := a j0x0 + · · · + a jn xn ∈HD[x0, . . . , xn] ( j = 1, . . . , 2t + 1). Define

D (H1, . . . , H2t+1) :=∏

L⊂{1,...,2t+1},#L=t+1

⎛⎝ ∑

{ j0,..., jn}⊂L

∣∣det (a j�i )0≤i,�≤n∣∣⎞⎠ .

The moving hyperplanes {Hj } are said to be in pointwise t−subgeneral position if forevery z ∈ D, the fixed hyperplanes {Hj (z)} are in t−subgeneral position. It is clear thatH1, . . . , H2t+1 are in pointwise t−subgeneral position in CPn if and only if D(H1, . . . ,

H2t+1)(z) > 0, for all z ∈ D.

For the case of hyperplanes, we obtain the following result.

Theorem 1.2 Let F be a family of meromorphic mappings of a domain D ⊂ Cm into

CPn . For each f in F , we consider 2t + 1 moving hyperplanes H1 f , · · · , H(2t+1) f inCPn such that {H∗

j f : f ∈ F} ( j = 1, . . . , 2t + 1) are normal families and there exists apositive constant δ0 satisfying

D(H1 f , . . . , H(2t+1) f )(z) > δ0, for all z ∈ D, f ∈ F .

Let m1, . . . , m2t+1 be positive integers and may be ∞ such that∑2t+1

j=11

m j< 1. Assume

that Hj f ( f ) �≡ 0 for all j ∈ {1, . . . , 2t + 1}, f ∈ F , and two following conditions aresatisfied:

(a) {z : 1 ≤ ν( f,Hj f )(z) ≤ m j } = {z : 1 ≤ ν(g,Hjg)(z) ≤ m j } for all f, g in F , and forall j ∈ {1, . . . , 2t + 1},

(b) I ( f ) ⊂ ∪2t+1j=1 {z : 1 ≤ ν( f,Hj f )(z) ≤ m j }, and Hj f ( f ) �≡ 0, for all j ∈

{1, . . . , 2t + 1} and f ∈ F , where I ( f ) is the set of all points of indeterminationof f.

Then F is a meromorphically normal family on D.

2. Notations

Let ν be a nonnegative divisor on C. For each positive integer (or +∞) p, we define thecounting function of ν (where multiplicities are truncated by p) by

N [p](r, ν) :=r∫

1

n[p]ν (t)

tdt (1 < r < +∞)

where n[p]ν (t) = ∑

|z|≤t min{ν(z), p}.

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4 N.T.T. Hang and T.V. Tan

For a holomorphic mapping f of C into CPn, and a homogeneous polynomial Q inCPn, with Q( f̃ ) �≡ 0, we define N [p]

( f,Q)(r) := N [p](r, ν( f,Q)). We simply write N( f,Q)(r)

for N [+∞]( f,Q) (r).

Let f be a holomorphic mapping of C into CPn . For arbitrary fixed homogeneouscoordinates (w0 : · · · : wn) of CPn , we take a reduced representation f̃ = ( f0, · · · , fn) off. Set ‖ f ‖ = max{| f0|, . . . , | fn|}.

The characteristic function of f is defined by

T f (r) := 1

2π

2π∫0

log‖ f (reiθ )‖dθ − 1

2π

2π∫0

log‖ f (eiθ )‖dθ, 1 < r < +∞.

We state the First and the Second Main Theorems in Nevanlinna theory:

First Main Theorem Let f be a holomorphic mapping of C into CPn and Q be ahomogeneous polynomial in C[x0, · · · , xn] of degree d ≥ 1, such that Q( f ) �≡ 0. Then

N( f,Q)(r) � d · T f (r) + O(1) for all r > 1.

Second Main Theorem Let f be a linearly nondegenerate holomorphic mappingof C into CPn and H1, . . . , Hq be fixed hyperplanes in CPn in N−subgeneral position(q > N ≥ n). Then,

(q − 2N + n − 1)T f (r) �q∑

j=1

N [n]( f,Hj )

(r) + o(T f (r)

)for all r except for a subset E of (1,+∞) of finite Lebesgue measure.

3. Proof of our results

In order to prove Theorems 1.1 and 1.2, we need some preparations.

Definition 3.1 ([4], Definition 4.4) Let {νi }∞i=1 be a sequence of non-negative divisors ona domain D in C

m . 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 nonzero holomorphic functions hand hi on U with νi = νhi and ν = νh on U such that {hi }∞i=1 converges to h uniformly oncompact subsets of U .

Let S be an analytic set in D of codimension ≥ 2. By Thullen-Remmert-Stein’s theorem[10], any nonnegative divisor ν on D\S can be uniquely extended to a divisor ν̂ on D.

Moreover, we have

Lemma 3.1 ([5], page 26, (2.9)) If a sequence {νk}∞k=1 of non-negative divisors on D\Sconverges to ν on D\S, then {̂νk} converges to ν̂ on D.

Lemma 3.2 ([5], Proposition 3.5) Let { fi } be a sequence of meromorphic mappings ofa domain D in C

m into CPn and let E be a thin analytic subset of D. Suppose that { fi }meromorphically converges on D\E to a meromorphic mapping f of D\E into CPn. If

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Complex Variables and Elliptic Equations 5

there exists a hyperplane H in CPn such that f (D\E) �⊂ H and {ν( fi ,H)} is a convergentsequence of divisors on D, then { fi } is meromorphically convergent on D.

Similarly to Corollary 3.1 in [5], we give the following corollary:

Corollary 3.1 Let S be an analytic set of codimension ≥ 2 in a domain D ⊂ Cm . Let

{ fk}∞k=1 be a sequence of meromorphic mappings of D into CPn . Assume that { fk |D\S}meromorphically converges to a meromorphic mapping f of D\S, then { fk} is meromor-phically convergent on D.

Proof Let H be a hyperplane in CPn such that f (D\S) �⊂ H. It is clear that {ν( fk ,H)} con-verges on D\S. Then, by Lemma 3.1, {ν( fk ,H)} converges on D. Therefore, byLemma 3.2, { fk} meromorphically converges on D. �

Lemma 3.3 ([8], Theorem 3.1) Let F be a family of holomorphic mappings of a domainD in C

m into CPn . The family F is not normal on D if and only if there exist sequences{p j } ⊂ D with {p j } → p0 ∈ D, { f j } ⊂ F , {ρ j } ⊂ R with ρ j > 0 and {ρ j } → 0, andEuclidean unit vectors {u j } ⊂ C

m, such that g j (ξ) := f j (p j + ρ j u jξ), where ξ ∈ C

satisfies p j + ρ j u jξ ∈ D, converges uniformly on compact subsets of C to a nonconstantholomorphic mapping g of C into CPn .

Lemma 3.4 ([2], Theorem 1) Let X be a closed subset of CPn (with respect to the usualtopology of a real manifold of dimension 2n) and let D1, . . . , D2�+1 be (fixed) hypersurfacesin CPn, in �−subgeneral position with respect to X. Then, every holomorphic mapping fof C into X\(∪2�+1

j=1 D j ) is constant.

Proof of Theorem 1.1 Assume that X is defined by homogeneous polynomialsQ2t+2, . . . , Qs in C[x0, . . . , xn]. By replacing Qi by Q

d jj where d j is a suitable positive

integer, we may assume that Q j ( j = 1, . . . , s) have the same degree d.

Set

Td := {(i0, . . . , in) ∈ N

n+10 : i0 + · · · + in = d

}.

Assume that

Q j =∑I∈Td

a j I x I ( j = 1, . . . , s)

where a j I ∈ HD for all I ∈ Td , j ∈ {1, . . . , 2t + 1}, a j I ∈ C for all I ∈ Td , j ∈{2t + 2, . . . , s}, x I = xi0

0 · · · xinn for x = (x0, . . . , xn) and I = (i0, . . . , in).

Let T = (. . . , tk I , . . . ) (k ∈ {1, . . . , s}, I ∈ Td ) be a family of variables.Set

Q̃ j =∑I∈Td

t j I x I ∈ Z[T, x], j = 1, . . . , s.

For each subset L ⊂ {1, · · · , 2t + 1} with |L| = t + 1, take R̃L ∈ Z[T ] is the resultantof (s − t) homogeneous polynomials Q̃ j , j ∈ {2t + 2, . . . , s} ∪ L . Since

{Q j

}j∈L are in

t−subgeneral position with respect to X , there exists z0 ∈ D such that (s − t) homogeneous

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6 N.T.T. Hang and T.V. Tan

polynomials Q2t+2, . . . , Qs, and Q j (z0), j ∈ L have nontrivial common solutions in Cn+1

(note that X is defined by the polynomials Qi ∈ C[x0, . . . , xn], i ∈ {2t + 2, . . . , s}). Thismeans that R̃L(. . . , ak I , . . . )(z0) �= 0.

By the assumption, for each j ∈ {1 . . . , 2t + 1}, the set A j := f −1(Q j ) does notdepend on the mapping f ∈ F .

Set

A :=⋃

L⊂{1,··· ,2t+1},#L=t+1

{z ∈ D : R̃L(· · · , ak I , · · · )(z) = 0}.

Then E := (∪2t+1i=1 Ai ) ∪ A is a thin analytic subset of D.

Let { fk}∞k=1 ⊂ F be an arbitrary sequence.For any fixed point z0 ∈ D\E , there exists an open ball B(z0, ε) in D\E such that

f −1k (Qi ) ∩ B(z0, ε) = ∅, for all k ≥ 1, and i ∈ {1, . . . , 2t + 1}. (3.1)

Since Qi ( f̃k) �= 0 on B(z0, ε), we get that B(z0, ε) ∩ I ( fk) = ∅. This implies,{ fk |B(z0,ε)}∞k=1 ⊂ Hol(B(z0, ε), CPm).

We now prove that { fk |B(z0,ε)}∞k=1 is a normal family on B(z0, ε). Indeed, suppose that{ fk |B(z0,ε)}∞k=1 is not normal on B(z0, ε), then by Lemma 3.3, there exist a subsequence(again denoted by { fk |B(z0,ε)}∞k=1) and p0 ∈ B(z0, ε), {pk}∞k=1 ∈ B(z0, ε) with pk →p0, {ρk} ⊂ (0,+∞) with ρ j → 0, Euclidean unit vectors {uk} ∈ C

m such that thesequence of holomorphic maps

gk(ξ) := fk(pk + ρkukξ) : rk → CPn, (rk → ∞)

converges uniformly on compact subsets of C to a nonconstant holomorphic mapg : C → CPn . Then, there exist reduced representations g̃k = (gk0, · · · , gkn) of gk anda representation g̃ = (g0, · · · , gn) of g such that {g̃ki } converges uniformly on compactsubsets of C to g̃i . This implies that Q j (pk + ρkukξ)(g̃k(ξ)) converges uniformly oncompact subsets of C to Q j (p0)(g̃(ξ)). By (3.1) and Hurwitz’s theorem, for each j ∈{1, . . . , 2t + 1} we have

Img ∩ Q j (p0) = ∅, or Img ⊂ Q j (p0).

Here, we identify the polynomial Q j (p0) ∈ C[x0, · · · , xn] with the hypersurface in CPn

defined by Q j (p0).

It is clear that Q1(p0), . . . , Q2t+1(p0) are in t−subgeneral position with respect to X,

sicne p0 �∈ A.

Since, Im fk ⊂ X, for all k ≥ 1, we get that Img ⊂ X . Without loss of generality, we mayassume that Img ⊂ Q j (p0) for 1 ≤ j ≤ v and Img∩ Q(p0) = ∅ for j ∈ {v+1, . . . , 2t +1}(we take v = 0 for the case where Img ∩ Q j (p0) = ∅ for all j ∈ {1, . . . , 2t + 1}).

We have Img ⊂ M := X ∩ (∩vj=1 Q j (p0)).

Case 1 v is even, v = 2�.

We consider 2(t−�)+1 hypersurfaces Qv+1(p0), . . . , Q2t+1(p0). For any subset T ⊂ {v+1, . . . , 2t +1}, with #T = (t −�)+1, we have M ∩(∩ j∈T Q j (p0)) = X ∩(∩ j∈T Q j (p0))∩(∩v

i=1 Qi (p0)) = ∅ (note that #(T ∪ {1, . . . , v}) = t − � + 1 + v = t + 1 + � ≥ t + 1).

This means that 2(t − �) + 1 hypersurfaces Qv+1(p0), . . . , Q2t+1(p0) are in (t − �)−subgeneral position with respect to M. On the other hand, Img ⊂ M\(∪2t+1

j=v+1 Q j (p0)).

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Complex Variables and Elliptic Equations 7

Hence, by Lemma 3.4, g is constant; this is a contradiction. Therefore, { fk |B(z0,ε)}∞k=1 is anormal family on B(z0, ε).

Case 2 v is old, v = 2� + 1.

We consider 2(t−�−1)+1 hypersurfaces Qv+1(p0), . . . , Q2t (p0). For any subset T ⊂ {v+1, . . . , 2t}, with #T = t − �, we have M ∩ (∩ j∈T Q j (p0)) = X ∩(∩ j∈T Q j (p0)) ∩ (∩v

i=1 Qi (p0)) = ∅ (note that #(T ∪ {1, . . . , v}) = (t − �) + v =t+1+� ≥ t+1). This means that 2(t−�−1)+1 hypersurfaces Qv+1(p0), . . . , Q2t (p0) arein (t − �)−subgeneral position with respect to M. On the other hand, Img ⊂M\(∪2t

j=v+1 Q j (p0)). Hence, by Lemma 3.4, g is constant; this is a contradiction. Hence,{ fk |B(z0,ε)}∞k=1 is a normal family on B(z0, ε).

By the usual diagonal argument, we can find a subsequence (again denoted by { fk}∞k=1)which converges uniformly on compact subsets of D\E to a holomorphic map f, Im f ⊂ X.

By the assumption, ∩2t+1j=1 A j is an analytic set of codimension ≥ 2. It is clear that

I ( fk) ⊂ A j for all k ≥ 1 and j ∈ {1, . . . , 2t + 1}. Therefore, I ( fk) ⊂ ∩2t+1j=1 A j for all

k ≥ 1 and j ∈ {1, . . . , 2t +1}. This means that { fk}k≥1 are holomorphic on D\(∩2t+1j=1 A j ).

For any fixed point z∗ ∈ D\(∩2t+1j=1 A j ), there exist an open ball B(z∗, ρ) ⊂ D\(∩2t+1

j=1 A j )

and an index j0 ∈ {1, . . . , 2t + 1} such that A j0 ∩ B(z∗, ρ) = ∅. This means that

f −1k (Q j0) ∩ B(z∗, ρ) = ∅. (3.2)

We define holomorphic mappings {Fk}∞k=1 of B(z∗, ρ) into CPn+1 as follows: for anyz ∈ B(z∗, ρ), if fk has a reduced representation f̃k = ( fk0, · · · , fkn) on a neighbourhoodUz ⊂ B(z∗, ρ) then Fk has a reduced representation F̃k = ( f d

k0, · · · , f dkn, Q j0( f̃k)) on Uz .

Let Hi (i = 0, · · · , n) be hyperplanes in CPn defined by

Hi = {(w0 : · · · wn)|wi = 0}and let Hi (i = 0, · · · , n + 1) be hyperplanes in CPn+1 defined by

Hi = {(w0 : · · · wn+1)|wi = 0}.It is easy to see that {Fk} converges uniformly on compact subset of B(z∗, ρ)\E to aholomorphic map F of B(z∗, ρ)\E into CPn+1, and if f has a reduced representationf̃ = ( f0, · · · , fn) on an open subset U ⊂ B(z∗, ρ)\E then F has reduced representationF̃ = ( f d

0 , · · · , f dn , Q j0( f̃ )) on U . Since f is holomorphic on B(z∗, ρ)\E , there exists

i0 (0 ≤ i0 ≤ n) such that Hi0(F̃) ≡ Hi0( f̃ ) �≡ 0 on B(z∗, ρ)\E . Then there exists k0 > 0such that Hi0(F̃k) ≡ Hi0( f̃k) �≡ 0 on B(z∗, ρ)\E for all k � k0.Since Q j0( f̃ ) �≡ 0 on B(z∗, ρ)\E , we have Hn+1(F̃) �≡ 0 on B(z∗, ρ)\E . On the otherhand, by (3.2), for all k ≥ 1, we have

F−1k (Hn+1) = f −1

k (Q j0) ∩ B(z∗, ρ) = ∅.

Therefore, by Lemma 3.2, {Fk} is meromorphically convergent on B(z∗, ρ). This impliesthat the sequence of divisors {ν(Fk ,Hi0 )} converges on B(z∗, ρ), and hence {ν( fk ,Hi0 )}converges on B(z∗, ρ). By again Lemma 3.2, { fk} meromorphically converges on B(z∗, ρ),

for any z∗ ∈ D\(∩2t+1j=1 A j ). Hence, { fk} meromorphically converges on D\(∩2t+1

j=1 A j ). On

the other hand, ∩2t+1j=1 A j is an analytic set of codimension ≥ 2. Hence, by Corollary 3.1,

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8 N.T.T. Hang and T.V. Tan

{ fk} meromorphically converges on D. Then F is a meromorphically normal family on D.We have completed the proof of Theorem 1.1. �

Proof of Theorem 1.2 Let { fk}∞k=1 ⊂ F be an arbitrary sequence. Since {H∗j f , f ∈ F}

is a (holomorphically) normal family, without loss of generality, we may assume that, foreach j ∈ {1, . . . , 2t + 1}, the sequence {H∗

j fk}∞k=1 converges uniformly on every compact

subset of D to a holomorphic map L∗j with the reduced representation (b j0 : · · · : b jn).

Let L j ( j = 1, . . . , 2t + 1) be moving hyperplanes defined by homogeneous polynomialsb j0x0 + · · · + b jn xn . Since

D(H1 fk , . . . , H(2t+1) fk )(z) ≥ δ0 for all z ∈ D, k ≥ 1,

we have D(L1, . . . , L2t+1)(z) ≥ δ0 for all z ∈ D. This means that L1, . . . , L2t+1 are inpointwise t−subgenral position in CPn .

By the assumption, for each j ∈ {1 . . . , 2t + 1}, the set S j := {z : 1 ≤ ν( f,Hj f ) ≤ m j }does not depend on the mapping f ∈ F .

We now prove that:

Claim 1 There exists a subsequence of { fk}∞k=1 (again denoted by { fk}∞k=1) thatconverges uniformly on compact subset of D\(∪2t+1

j=1 S j ) to a holomorphic map f on

D\(∪2t+1j=1 S j ). Here S j is the colosure of S j .

For any fixed point z0 ∈ D\(∪2t+1j=1 S j ), we take an open ball B(z0, ε) ⊂ D\(∪2t+1

j=1 S j ).

Then, for all j ∈ {1, . . . , 2t + 1},S j ∩ B(z0, ε) = ∅. (3.3)

This implies, Hj fk ( fk) has no zeros of multiplicity ≥ m j on B(z0, ε) for all k ≥ 1 andj ∈ {1, . . . , 2t + 1}. Since I ( fk) ⊂ ∪2t+1

j=1 S j , we have that { fk}∞k=1 are holomorphic onB(z0, ε).

We now prove that { fk |B(z0,ε)}∞k=1 is a normal family on B(z0, ε). Indeed, suppose that{ fk |B(z0,ε)}∞k=1 is not normal on B(z0, ε), then by Lemma 3.3, there exist a subsequence(again denoted by { fk |B(z0,ε)}∞k=1) and p0 ∈ B(z0, ε), {pk}∞k=1 ∈ B(z0, ε) with pk →p0, {ρk} ⊂ (0,+∞) with ρ j → 0, Euclidean unit vectors {uk} ∈ C

m such that thesequence of holomorphic maps

gk(ξ) := fk(pk + ρkukξ) : rk → CPn, (rk → ∞)

converges uniformly on compact subsets of C to a nonconstant holomorphic map g : C →CPn . Then, there exist reduced representations g̃k = (gk0, · · · , gkn) of gk and a reducedrepresentation g̃ = (g0, · · · , gn) of g such that {g̃ki } converges uniformly on compactsubsets of C to g̃i . This implies that Hj (pk+ρkukξ)(g̃k(z)) converges uniformly on compactsubsets of C to L j (p0)(g̃(ξ)). By (3.3) and Hurwitz’s theorem, for each j ∈ {1, . . . , 2t +1}we have that Img ⊂ L j (p0), or L j (p0)(g̃) have no zeros of multiplicity ≤ m j .

Since L1, . . . , L2t+1 are in pointwise t−subgeneral position in CPn, we have that fixedhyperplanes L1(p0), . . . , L2t+1(p0) are in t−subgeneral position in CPn .

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Complex Variables and Elliptic Equations 9

Case 1 Img �⊂ L j (p0), for all j ∈ {1, . . . , 2t + 1}. Let CP� ⊂ CPn be the small leastprojective subspace satisfying Img ⊂ CP�.Then CP� �⊂ L j (p0) for all j ∈ {1, . . . , 2t+1},and (fixed) hyperplanes L1(p0)∩CP�, . . . , L2t+1(p0)∩CP� are in t−subgeneral positionin CP�. Since g is nonconstant, we have � ≥ 1. It is clear that g is linearly nondegeneratein CP�. By the First and Second Main Theorems, we have

� · Tg(r) = (2t + 1 − 2t + � − 1)Tg(r)

≤2t+1∑j=1

N [�](g,L j (p0))

(r) + o(Tg(r))

≤2t+1∑j=1

�

m jN(g,L j (p0))(r) + o(Tg(r))

≤2t+1∑j=1

�

m jTg(r) + o(Tg(r)),

for all r > 1 except for a subset of (1,+∞) of finite Lebesgue measure. Therefore,∑2t+1j=1

1m j

≥ 1; this is a contradiction.

Case 2 There exists j ∈ {1, . . . , 2t + 1} such that Img ⊂ L j (p0). Without loss ofgenerality, we may assume t hat Img ⊂ L j (p0) for j ≤ k (k ≥ 1), and Img �⊂ L j (p0) forj ∈ {k + 1, . . . , 2t + 1}.

Set P := (∩kj=1L j (p0)). Let CPs ⊂ P be the small least projective subspace containing

Img. Since g is nonconstant, we get that s ≥ 1. Since, L1(p0), . . . , L2t+1(p0) are int−subgeneral position in CPn, we have that (2t − k + 1) (fixed) hyperplanes Lk+1(p0) ∩CPs, . . . , L2t+1(p0) ∩ CPs are in (t − k) subgeneral position in CPs (note that CPs �⊂L j (p0) for all j ∈ {k + 1, . . . , 2t + 1}). Therefore, by the First and Second Main Theorem,for any ε′ > 0, we have

(k + s)Tg(r) = (2t − k + 1 − 2(t − k) + s − 1)Tg(r)

≤2t+1∑

j=k+1

N [s](g,L j (p0))

(r) + o(Tg(r))

≤2t+1∑

j=k+1

s

m jN(g,L j (p0))(r) + o(Tg(r))

≤2t+1∑

j=k+1

s

m jTg(r) + o(Tg(r)),

for all r > 1 except for a subset of (1,+∞) of finite Lebesgue measure. Therefore,∑2t+1j=k+1

1m j

≥ k+ss > 1; this is a contradiction.

Hence, { fk |B(z0,ε)}∞k=k0is a normal family on B(z0, ε). By the usual diagonal argument,

we can find a subsequence (again denoted by { fk}∞k=1) which converges uniformly oncompact subset of D\(∪2t+1

j=1 S j ) to a holomorphic map f. The proof of Claim 1 has beencompleted. �

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10 N.T.T. Hang and T.V. Tan

Take a fixed mapping f∗ ∈ F with a reduced representation f̃∗. Set

S◦j := {z ∈ S j : z is a regular point of Hj f∗( f̃∗)}, j = 1, . . . , 2t + 1.

Under the notation as in Claim 1, we now prove that:

Claim 2 For each z∗ ∈ ∪2t+1j=1 S◦

j , there exists an open ball B(z∗, ε) such that { fk}meromorphically converges on B(z∗, ε).

Let an arbitrary point z∗ ∈ S◦j0, for some j0.

We have m∗j0

:= ν( f∗,Hj0 f∗ )(z∗) ≤ m j0 (note that z∗ ∈ S j0). Since z∗ is a regular point

of the function Hj0 f∗( f̃∗), there exists a ball B(z∗, ε) ⊂ D such that all the zero points ofHj0 f∗( f̃∗) in B(z∗, ε) have the same multiplicity m∗

j0. Therefore, Hj0 f ( f̃ ) ( f ∈ F have the

same zero set in B(z∗, ε) and all the zero points of Hj0 f ( f̃ ) in B(z∗, ε) have multiplicitym∗

j0. Therefore,

{ν( fk ,Hj0 fk )|B(z∗,ε)}∞k=1 is a constant sequence of divisors. (3.4)

We define meromorphic mappings {gk}∞k=1 of B(z∗, ε) into CPn+1 as follows: for anyz ∈ B(z∗, ε), if fk has a reduced representation f̃k = ( fk0, · · · , fkn) on a neighbourhoodUz ⊂ B(z∗, ε) then gk has a reduced representation g̃k = ( f d

k0, · · · , f dkn, Hj0 fk ( f̃k)) on Uz .

Let Hi (i = 0, · · · , n) be hyperplanes in CPn defined by

Hi = {(w0 : · · · wn)|wi = 0}and let Hi (i = 0, · · · , n + 1) be hyperplanes in CPn+1 defined by

Hi = {(w0 : · · · wn+1)|wi = 0}.Set E := ∪2t+1

j=1 f −1∗ (Hj f ∗). Then E is a thin analytic subset of D and ∪2t+1j=1 S j ⊂ ∪2t+1

j=1f −1∗ (Hj f ∗). By Claim 1, it is easy to see that {gk} converges uniformly on compact subsetsof B(z∗, ρ)\E to a holomorphic map g of B(z∗, ε)\E into CPn+1, and if f has a reducedrepresentation f̃ = ( f0, · · · , fn) on an open subset U ⊂ B(z∗, ρ)\E then g has reducedrepresentation g̃ = ( f d

0 , · · · , f dn , L j0( f̃ )) on U . Since f is holomorphic on B(z∗, ε)\E ,

there exists i0 (0 ≤ i0 ≤ n) such that Hi0( f̃ ) �≡ 0 on B(z∗, ε)\E , and hence Hi0(g̃) �≡ 0on B(z∗, ε)\E . Then, there exists k0 > 0 such that Hi0( f̃k) ≡ Hi0(g̃k) �≡ 0 on B(z∗, ε)\Efor all k � k0.

We have ν(gk ,Hn+1)≡ ν( fk ,Hj0 fk ) on B(z∗, ε). Therefore, by (3.3) and Lemma 3.3, {gk}

meromorphically converges on B(z∗, ε). It implies that {ν(gk ,Hi0 )} converges on B(z∗, ρ),and hence {ν( fk ,Hi0 )} converges on B(z∗, ρ). By again Lemma 3.3, we get that { fk}k≥k0

meromorphically converges on B(z∗, ε). This completes the proof of Claim 2. �We have S j\S◦

j ⊂ sing f −1∗ (Hj f∗), where sing f −1∗ (Hj f∗) means the singular locus ofthe (reduction of the) analytic set f −1∗ (Hj f∗). Indeed, otherwise there existed a ∈ (S j\S◦

j )∩reg f −1∗ (Hj f∗).Then m∗ := ν( f∗,Hj f∗ )(a) > m j . Since a is a regular point of ( f −1∗ (Hj f∗), byRückert Nullstellensatz, there exists nonzero holomorphic function h, u on a neighbourhoodU of a such that dh and u have no zero point and Hj f∗( f∗) = hm∗ · u on U. Since a ∈ S j ,

there exists b ∈ S j ∩ U. Hence, m j ≥ ν( f∗,Hj f∗ )(b) = m∗; this is a contradiction. Thus,S j\S◦

j ⊂ sing f −1∗ (Hj f∗). Set S := ∪2t+1j=1 sing f −1∗ (Hj f∗). Then S is an analytic set in D of

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Complex Variables and Elliptic Equations 11

codimension ≥ 2. We have ∪2t+1j=1 (S j\S◦

j ) ⊂ S. Therefore, from Claims 1 and 2, there existsa subsequence of { fk}∞k=1 (again denoted by { fk}∞k=1) which meromorphically convergeson D\S. Hence, by Corollary 3.1, { fk}∞k=1 meromorphically converges on D. Hence, F isa meromorphically normal family on D. We have completed the proof of Theorem 1.2. �

AcknowledgementWe thank Si Duc Quang for helpful discussions.

FundingThis research is funded by Vietnam National Foundation for Science and Technology Development(NAFOSTED). The second named author was partially supported by Vietnam Institute for AdvancedStudy in Mathematics, and by the Abdus Salam International Centre for Theoretical Physics, Italy.

References

[1] Green M. Holomorphic maps into complex projective space omiting hyperplanes. Trans. Am.Math. Soc. 1972;169:89–103.

[2] Eremenko A. A Picard type theorem for holomorphic curves. Period. Math. Hung. 1999;38:39–42.

[3] Tu Z. Normal criteria for families of holomorphic mappings of several complex variables intoCPn . Proc. Amer. Math. Soc. 2005;127:1039–1049.

[4] Tu Z, Li P. Normal families of meromorphic mappings of several complex variables into CPn

for moving targets. Sci. China Ser. A Math. 2005;48:355–368.[5] Fujimoto H. On families of meromorphic maps into the complex projective space. Nagoya Math.

J. 1974;54:21–51.[6] Dethloff G, Thai DD, Trang PNT. Normal families of meromorphic mappings of several

complex variables for moving hypersurfaces in a complex projective space. Preprint math.CV/arXiv:13010687.

[7] Quang SD, Tan TV. Normal families of meromorphic mappings of several complex variablesinto CPn for moving hypersurfaces. Ann. Polon. Math. 2008;94:97–110.

[8] Mai PN, Thai DD, Trang PNT. Normal families of meromorphic mappings of several complexvariables into CPn . Nagoya Math. J. 2005;180:91–110.

[9] Thullen P. Über die wesentlichen Singularitäten analytischer Functionen und Fláchen in Raumvon n komplexen Veränderlichen [On essential singularities of analytic functions and surfacesin Cn]. Math. Ann. 1935;111:137–157.

[10] Aladro G, Krantz SG. A criterion for normality in Cn, J Math. Anal. Appl. 1991;161:1–8.

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