Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C)
We present some extension theorems in the style of the Big Picard theorem for meromorphic mappings of Cm into Pn(C) with a few moving hyperplanes.
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Інститут математики НАН України
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irk-123456789-1663132020-02-19T01:27:48Z Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) Quang, Si Duc Giang, Ha Huong Статті We present some extension theorems in the style of the Big Picard theorem for meromorphic mappings of Cm into Pn(C) with a few moving hyperplanes. Наведено дєякі теореми про продовження в стилі великої теореми Шкара для мероморфних відображень Cm в Pn(C) з деякими рухомими гіперплощинами. 2014 Article Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) / Si Duc Quang, Ha Huong Giang // Український математичний журнал. — 2014. — Т. 66, № 11. — С. 1550–1562. — Бібліогр.: 11 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166313 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Quang, Si Duc Giang, Ha Huong Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) Український математичний журнал |
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We present some extension theorems in the style of the Big Picard theorem for meromorphic mappings of Cm into Pn(C) with a few moving hyperplanes. |
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Article |
| author |
Quang, Si Duc Giang, Ha Huong |
| author_facet |
Quang, Si Duc Giang, Ha Huong |
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Quang, Si Duc |
| title |
Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) |
| title_short |
Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) |
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Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) |
| title_fullStr |
Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) |
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Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) |
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big picard theorem for meromorphic mappings with moving hyperplanes in pn(c) |
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Інститут математики НАН України |
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2014 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166313 |
| citation_txt |
Big Picard theorem for meromorphic mappings with moving hyperplanes in Pn(C) / Si Duc Quang, Ha Huong Giang // Український математичний журнал. — 2014. — Т. 66, № 11. — С. 1550–1562. — Бібліогр.: 11 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT quangsiduc bigpicardtheoremformeromorphicmappingswithmovinghyperplanesinpnc AT gianghahuong bigpicardtheoremformeromorphicmappingswithmovinghyperplanesinpnc |
| first_indexed |
2025-07-14T21:08:28Z |
| last_indexed |
2025-07-14T21:08:28Z |
| _version_ |
1837658075582955520 |
| fulltext |
UDC 517.9
Si Duc Quang (Hanoi Nat. Univ. Education, Vietnam),
Ha Huong Giang (Electric Power Univ., Hanoi, Vietnam)
BIG PICARD THEOREM FOR MEROMORPHIC MAPPINGS
WITH MOVING HYPERPLANES IN Pn(C)*
ВЕЛИКА ТЕОРЕМА ПIКАРА ДЛЯ МЕРОМОРФНИХ ВIДОБРАЖЕНЬ
З РУХОМИМИ ГIПЕРПЛОЩИНАМИ В Pn(C)
We give some extension theorems in the style of Big Picard theorem for meromorphic mappings of Cm into Pn(C) with
a few moving hyperplanes.
Наведено деякi теореми про продовження в стилi великої теореми Пiкара для мероморфних вiдображень Cm в
Pn(C) з деякими рухомими гiперплощинами.
1. Introduction and main results. As well known, in complex one variable, Picard proved the
following theorems for meromorphic functions.
Theorem A (Little Picard theorem). Let f(z) be a meromorphic function on the complex plane.
If there exist three mutually distinct points w1, w2 and w3 on the Riemann sphere such that f(z)−wi,
i = 1, 2, 3, has no zero on the complex plane, then f is a constant.
Theorem B (Big Picard theorem). Let f(z) be a meromorphic function on ∆∗ = {z ∈ C :
1 ≤ |z| < +∞}. If there exist three mutually distinct points w1, w2 and w3 on the Riemann sphere
such that f(z) − wi, i = 1, 2, 3, has no zero on ∆∗, then f does not have an essential singularity
at∞.
In the case of higher dimension, H. Fujimoto [3] improved Theorem B as follows.
Theorem C ([3], Theorem A). LetM be a complex manifold and let S be a regular thin analytic
subset of M and let f be a holomorphic map of M \ S into the n-dimensional complex projective
space Pn(C). If f is of rank r somewhere and if f(M \ S) omits 2n− r+ 2 hyperplanes in general
position, then f can be extended to a holomorphic map of M into Pn(C), where the rank of f at a
point x ∈M \ S means the rank of the Jacobian matrix of f at x.
In 2006, by using a criterion on normality and applying little Picard theorems for holomorphic
mappings, Z. H. Tu generalized Big Picard’s theorem to the case of moving hyperplanes as follows.
Theorem D ([11], Theorem 2.2). Let S be an analytic subset of a domain D in Cn with codi-
mension one, whose singularities are normal crossings. Let f be a holomorphic mapping from D \S
into Pn(C). Let a1(z), . . . , aq(z), z ∈ D, be q, q ≥ 2n+ 1, moving hyperplanes in Pn(C) located
in pointwise general position such that f(z) intersects aj(z) on D \ S with multiplicity at least mj ,
j = 1, . . . , q, where m1, . . . ,mq are positive integers and may be +∞, with
q∑
j=1
1
mj
<
q − (n+ 1)
n
.
Then the holomorphic mapping f from D \ S into Pn(C) extends to a holomorphic mapping from
D into Pn(C).
* The research of the first author was supported in part by a NAFOSTED grant of Vietnam.
c© SI DUC QUANG, HA HUONG GIANG, 2014
1550 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
BIG PICARD THEOREM FOR MEROMORPHIC MAPPINGS WITH MOVING HYPERPLANES IN Pn(C) 1551
We would like to note that in Theorem D, the number q of moving hyperplanes is assumed to be
at least 2n+ 1 and the technique of its proof does not work in the case where q < 2n+ 1. Then the
natural question arise here:
Are there any extension theorem which is similar to Theorem D for the case where the number of
moving hyperplanes is less than 2n+ 1?
In this paper, we will give some positive answers for this question.
Firstly, we recall some notions due to [8, 10, 11].
Let D be a domain in Cm. We mean a moving hyperplane of Pn(C) on D a holomorphic
mapping a from D into Pn(C) with a reduced representation a = (a0 : . . . : an), where a0, . . . , an
are holomorphic functions on D without common zeros. Sometime we regard a(z) as a hyperplane
a(z) =
{
(ω0 : . . . : ωn) ∈ Pn(C) :
∑n
j=0
aj(z)ωj = 0
}
.
Let f be a meromorphic mapping of D into Pn(C). Denote by Df the smallest linear subspace
of Pn(C) which contains f(D) and denote by Lf the dimension of Df . For z ∈ D, take a reduced
representation f = (f0 : . . . : fn) of f on a neighborhood Uz of z and set (f, a) :=
∑n
j=0
ajfj on Uz.
We define div(f, a)(z) := div
(∑n
j=0
ajfj
)
(z) if (f, a) 6≡ 0 and div(f, a)(z) := ∞ if (f, a) ≡ 0.
Thus, div(f, a) is well-defined on D independently of the choice of reduced representations of f. If
div(f, a)(z) ≥ mj for all z ∈ D, we say that f intersects a on D with multiplicity at least mj .
Let A = {a0, . . . , aq−1} be a set of q moving hyperplanes of Pn(C) on D. Assume that each ai
has a reduced representation ai = (ai0 : . . . : ain). Denote byR{ai} the smallest field which contains
C and all functions
aik
aij
with aij 6≡ 0. Sometime we write R for R{ai} if there is no confusion and
denote by (A)R the linear span of A over R. We say that:
A is located in general position on D if and only if for any arbitrary n + 1 moving hyperlanes
{aik}1≤k≤n+1 ⊂ A there exists a point z ∈ D such that ∩1≤k≤n+1aik(z) = ∅.
A is located in pointwise N -subgeneral position on D if and only if for any arbitrary N + 1
moving hyperplanes {aik}1≤k≤N+1 ⊂ A then ∩1≤k≤N+1aik(z) = ∅ for all z ∈ D.
A is located in pointwise N -subgeneral position on D with respect to f if and only if for any
arbitrary N + 1 moving hyperlanes {aik}
N+1
k=1 ⊂ A then ∩N+1
k=1 aik(z) ∩Df = ∅ for all z ∈ D.
Then we see that if A is located in pointwise N -subgeneral position on D then it will be located
in pointwise N -subgeneral position on D with repect to f for every mapping f, but not vice versa.
Our main result of this work is stated as follows.
Theorem 1.1. Let f be a holomorphic mapping of a domain D \ S into Pn(C), where D is a
domain in Cm and S is an analytic subset of codimension one ofD. Let a1, . . . , an+2 be n+2 moving
hyperplanes in Pn(C) on D located in general position so that f is linearly nondegenerate over
R{ai}. Assume that f intersects each ai on D \S with multiplicity at least mi, where m1, . . . ,mn+2
are fixed positive integers and may be +∞, with
n+2∑
i=1
1
mi
<
1
n
.
Then f extends to a meromorphic mapping f̃ from D into Pn(C).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
1552 SI DUC QUANG, HA HUONG GIANG
In the last section of this paper, we will consider the case where the condition linearly nondegeneracy
of mappings is omitted.
2. Basic notions and auxiliary results from Nevanlinna theory. 2.1. We set punctured discs
on Ĉ = C ∪ {∞} about ∞ by
∆∗ = {z ∈ C : |z| ≥ 1},
∆∗(t) = {z ∈ C : |z| ≥ t}, t ≥ 1,
and we set
Γ(t) = {z ∈ C : |z| = t}, t ≥ 1.
In this paper, we always assume that functions on ∆∗ and mappings from ∆∗ are defined on a
neighborhood of ∆∗ in C. Let ξ be a function on ∆∗ satisfying that
(i) ξ is differentiable outside a discrete set of points,
(ii) ξ is locally written as a difference of two subharmonic functions.
Then by [7] (§ 1), we have
t∫
1
dt
t
∫
∆∗(t)
ddcξ =
1
4π
∫
Γ(r)
ξ(reiθ)dθ − 1
4π
∫
Γ(1)
ξ(reiθ)dθ − (log r)
∫
Γ(1)
dcξ, (2.1)
where ddcξ is taken in the sense of current.
2.2. A divisor E on ∆∗ is given by a formal sum E =
∑
µνpν , with {pν} is a locally
finite family of distinct points in ∆∗ and µν ∈ Z. We define the support of the divisor E by
Supp (E) =
⋃
µν 6=0 pν . Let k be a positive integer or +∞. We define the divisor E(k) by
E(k) :=
∑
min{µν , k}pν
and the truncated counting function to level k of E by
N (k)(r, E) :=
r∫
1
n(k)(t, E)
t
dt, 1 < r < +∞,
where
n(k)(t, E) =
∑
|z|≤t
E(k)(z).
We simply write N(r, E) for N (+∞)(r, E).
2.3. Let f : ∆∗ → Pn(C) be a holomorphic curve. For an arbitrary fixed homogeneous
coordinates (w0 : . . . : wn) of Pn(C), there exist a neighborhood U of ∆∗ in Cm and a reduced
representation (f0 : . . . : fn) on U of f, which means that f0, . . . , fn are holomorphic functions on
U without common zeros. We set ‖f‖ :=
(
|f0|2 + . . .+ |fn|2
)1/2
.
Denote by Ω the Fubini – Study form of Pn(C). The order function or characteristic function of
f with respect to Ω is defined by
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
BIG PICARD THEOREM FOR MEROMORPHIC MAPPINGS WITH MOVING HYPERPLANES IN Pn(C) 1553
Tf (r) := Tf (r; Ω) =
r∫
1
dt
t
∫
∆∗(t)
f∗Ω, r > 1. (2.2)
Applying (2.1) to ξ = log ‖f‖, we obtain
Tf (r) =
1
2π
∫
Γ(r)
log ‖f(reiθ)‖dθ − 1
2π
∫
Γ(1)
log ‖f(eiθ)‖dθ − (log r)
∫
Γ(1)
dc log ‖f‖. (2.3)
Let a be a moving hyperplane in Pn(C) with a reduced representation a = (a0 : . . . : an). We
set (f, a) =
∑n
i=0
aifi. Assume that (f, a) 6≡ 0, we define the proximity function of f with respect
to a by
mf (r, a) =
1
2π
∫
Γ(r)
log
‖f‖‖a‖
|(f, a)|
dθ − 1
2π
∫
Γ(1)
log
‖f‖‖a‖
|(f, a)|
dθ,
where ‖a‖ =
(∑n
i=0
|ai|2
)1/2
.
Applying (2.1) to ξ = log |(f, a)|, we get
N(r, div(f, a)) =
1
2π
∫
Γ(r)
log |(f, a)|dθ − 1
2π
∫
Γ(1)
log |(f, a)|dθ − (log r)
∫
Γ(1)
dc log |(f, a)|. (2.4)
Combining (2.2) and (2.4), we have the First Main Theorem as follows:
Tf (r) + Ta(r) = N(r, div(f, a)) +mf (r, a) + (log r)
∫
Γ(1)
dc log
(
‖f‖ ‖a‖
|(f, a)|
)
. (2.5)
2.4. For a meromorphic function ϕ on ∆∗, applying (2.1) to ξ = log |ϕ|, we obtain
N(r, div0(ϕ)) +N(r, div∞(ϕ)) =
=
1
2π
∫
Γ(r)
log |ϕ|dθ − 1
2π
∫
Γ(1)
log |ϕ|dθ − (log r)
∫
Γ(1)
dc log |ϕ|.
The proximity function m(r, ϕ) is defined by
m(r, ϕ) :=
1
2π
∫
Γ(r)
log+ |ϕ|dθ,
where log+ x = max
{
log x, 0
}
for x > 0. The Nevanlinna’s characteristic function is defined by
T (r, ϕ) := N(r, div∞(ϕ)) +m(r, ϕ).
We regard ϕ as a meromorphic mapping from ∆∗ into P1(C). There is a fact that
Tϕ(r) = T (r, ϕ) +O(log r).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
1554 SI DUC QUANG, HA HUONG GIANG
Theorem 2.1 (Lemma on logarithmic derivative [7]). Let ϕ be a nonzero meromorphic function
on ∆∗. Then ∣∣∣∣∣∣∣∣ m
(
r,
ϕ′
ϕ
)
= O(log+ Tϕ(r)) + C log r, (2.6)
where C is a positive constant which does not depend on ϕ.
As usual, by the notation “‖ P ” we mean the assertion P holds for all r ∈ (1,+∞) excluding a
finite Lebesgue measure subset E of (1,+∞).
3. Extension of meromorphic mappings with (n + 2) moving hyperplanes. In this section,
we will give the proof of Theorem 1.1. We need the following lemmas.
Firstly, we know the following characterization of a removable singularity, a classical result of
J. Noguchi (cf. [7]).
Lemma 3.1. Let f : ∆∗ → Pn(C) be a holomorphic curve. Then f extends at ∞ to a holo-
morphic curve f̃ from ∆ = ∆∗
⋃
{∞} into Pn(C) if and only if
lim inf
r→∞
Tf (r)/(log r) <∞.
The following lemma is due to Brownawell and Masser [2].
Lemma 3.2. Assume
∑n+1
i=0
fi = 0 and
∑
i∈I
fi 6= 0 for every I {0, . . . , n + 1}. Then we
can find a partition
{f0, . . . , fn+1} = A1
⋃
A2
⋃
. . .
⋃
Ak, k ≥ 1,
into nonempty disjoint sets A1, . . . , Ak, and nonempty sets A′1 ⊂ A1, A
′
2 ⊂ A1
⋃
A2, . . . , A
′
k−1 ⊂
⊂ A1
⋃
. . .
⋃
Ak−1 such that A1, A2
⋃
A′1, . . . , Ak
⋃
A′k−1 are minimal. Here, we say that a subset
A of {f0, . . . , fn+1} is minimal if it is linearly dependent, and any its proper subset is linearly
independent.
We now prove a Second Main Theorem for meromorphic mappings from punctured disks with
moving hyperplanes as follows.
Lemma 3.3. Let f be a holomorphic curve from the punctured disc ∆∗ into Pn(C) with a
reduced representation f = (f0 : . . . : fn), and let fn+1 = −f0 − . . .− fn so that∑
i∈I
fi 6= 0 ∀I ( {0, . . . , n+ 1}.
Then the following holds:∥∥∥∥ Tf (r) ≤
n+1∑
i=0
N (n)(r, div0(fi)) +O(log+ Tf (r)) +O(log r).
Proof. Set A = {f0, . . . , fn+1}. By the assumption, then there exist a partition A = A1
⋃
. . .
. . .
⋃
Ak and nonempty subsets A′s, 1 ≤ s ≤ k − 1, as in Lemma 3.3. By changing indices if
necessary, we may assume that
A1 = {0, 1, . . . , t1}, As = {ts−1 + 1, ts−1 + 2, . . . , ts}, t0 = 0, tk = n+ 1, 2 ≤ s ≤ k.
Since A1 is minimal, there exist nonzero constants c1i, 0 ≤ i ≤ t1, so that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
BIG PICARD THEOREM FOR MEROMORPHIC MAPPINGS WITH MOVING HYPERPLANES IN Pn(C) 1555
t1∑
i=0
α1ifi = 0.
Similarly, for s > 1, since As
⋃
A′s−1 is minimal, there exist nonzero constants csi, ts−1 < i ≤ ts,
and constants csi, 0 ≤ i ≤ ts−1, so that
ts∑
i=0
αsifi = 0.
We set csi = 0 for all i > ts, s ≥ 1. Then we have
k∑
s=1
ts∑
i=ts−1+1
αsifi = 0. (3.1)
Since A1 \ {0} and As, s ≥ 2, are linearly independent, then
Ds = det
(
Dl(αsifi); 0 ≤ l ≤ ts − ts−1 − 1, ts−1 + 1 ≤ i ≤ ts
)
6= 0,
where Dl denotes the derivatives of order l. Consider an minor (n+ 1)× (n+ 2)-matrices T and T̃
given by
T =
(
Dl(αsifi)
)
0≤l≤ts−ts−1−1, 1≤s≤k
0≤i≤n+1
,
T̃ =
(
Dl
(
αsifi
f0
))
0≤l≤ts−ts−1−1, 1≤s≤k
0≤i≤n+1
.
Denote by Bi (resp. B̃i) the determinant of the matrix obtained by deleting the (i+ 1)-th column of
the minor matrix T (resp. T̃ ). Since the sum of each row of T (resp. T̃ ) is zero, we actually have
Bi = (−1)iB0 = (−1)i
k∏
i=1
Di = (−1)ifn+1
0
k∏
i=1
D̃i = (−1)ifn+1
0 B̃0 = fn+1
0 B̃i.
We see that there exists a constant C > 0 so that ‖f(z)‖ ≤ C : max{|f0(z)|, . . . , |fn+1(z)|} for all
z ∈ ∆∗. Therefore, we get
‖f(z)‖ |B0(z)|∏n+1
i=0
|fi(z)|
≤ C max{|f0(z)|, . . . , |fn+1(z)|}|B0(z)|∏n+1
i=0
|fi(z)|
.
This yields that
log ‖f(z)‖+ log
|B0(z)|∏n+1
i=0
|fi(z)|
≤
n+1∑
i=0
log+ |Bi(z)|∏n+1
j=0,j 6=i
|fj(z)|
+O(1) =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
1556 SI DUC QUANG, HA HUONG GIANG
=
n+1∑
i=0
log+ |B̃i(z)|∏n+1
j=0,j 6=i
∣∣∣∣fjf0
(z)
∣∣∣∣ +O(1).
Integrating both sides of this inequality over Γ(r) and applying the lemma on logarithmic derivatives,
we obtain ∥∥∥∥ Tf (r) +
1
2π
∫
Γ(r)
log
|B0|∏n+1
i=0
|fi|
dθ ≤ O(log+ Tf (r)) + C1 log r.
Thus
‖ Tf (r) ≤
k∑
s=1
∑
fi∈As
N(r, div(fi))−N(r, div(As))
+O(log+ Tf (r)) + C1 log r ≤
≤
k∑
i=1
∑
fi∈As
N ts−ts−1−1(r, div(fi)) +O(log+ Tf (r)) + C1 log r ≤
≤
n+1∑
i=0
N (n)(r, div(fi)) +O(log+ Tf (r)) + C1 log r.
The lemma is proved.
Lemma 3.4. Let f be a holomorphic curve from a punctured disc ∆∗ into Pn(C), and let
a1, . . . , an+2 be n+ 2 moving hyperplanes in Pn(C) on ∆ located in general position so that there
exist nonzero moromorphic functions αi, 1 ≤ i ≤ n+ 2, on ∆ satisfying:
∑n+2
i=1
αi(f, ai) = 0 and∑
i∈I
αi(f, ai) 6= 0 ∀I {1, . . . , n+ 2}.
Assume that f intersects each ai on ∆∗ with multiplicity at least mi, where m1, . . . ,mn+2 are fixed
integers and may be +∞, with
n∑
i=1
1
mi
<
1
n
.
Then f extends at∞ to a holomorphic curve f̃ from ∆ = ∆∗
⋃
{∞} to Pn(C).
Proof. Without loss of generality, we may assume that αi, 1 ≤ i ≤ n + 2, have no neither
common zero nor pole. We consider the following divisor on ∆∗ as follows:
ν(z) = min
{
div(αi(f, ai))(z); 1 ≤ i ≤ n+ 2
}
.
Since f is holomorphic, it easy to see that Supp (ν) is subset of
⋃
1≤i0<...<in≤n+2
{z | rankC(ai0(z), . . . , ain(z)) ≤ n}
⋃ n+1⋃
i=1
Supp (div(αi)),
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BIG PICARD THEOREM FOR MEROMORPHIC MAPPINGS WITH MOVING HYPERPLANES IN Pn(C) 1557
which is an analytic subset of ∆. Therefore we may consider ν as a divisor on ∆. Choose a
holomorphic function h on Cm so that div(h) = ν and set Fi =
1
h
αi(f, ai). Then we see that∑n+2
i=1
Fi = 0, (F1 : . . . : Fn+1) is a reduced representation of a holomorphic curve F and∑
i∈I
Fi 6≡ 0 ∀I {1, . . . , n+ 2}.
Hence F satisfy the assumption of Lemma 3.3, then
‖ TF (r) ≤
n+2∑
i=1
N (n)
(
r, div0
(αi
h
(f, ai)
))
+O(log+ TF (r)) +O(log r) =
=
n+2∑
i=1
N (n)(r, div0(f, ai))) +O
(
n+2∑
i=1
T
(
r,
αi
h
))
+O(log+ TF (r)) +O(log r) ≤
≤
n+2∑
i=1
n
mi
N(r, div0(f, ai)) +O
(
n+2∑
i=1
T
(
r,
αi
h
))
+O(log+ TF (r)) +O(log r) ≤
≤
(
n+2∑
i=1
n
mi
)
Tf (r) +O
(
n+2∑
i=1
Tai(r)
)
+O
(
n+2∑
i=1
T
(
r,
αi
h
))
+O(log+ TF (r)) +O(log r).
Since ai,
αi
h
, 1 ≤ i ≤ n+ 2, are holomorphic on ∆, then by Lemma 3.2 we have
n+2∑
i=1
(
Tai(r) + T
(
r,
αi
h
))
= O(log r).
Thus
‖ TF (r) ≤
(
n+2∑
i=1
n
mi
)
Tf (r) +O(log+ TF (r)) +O(log r).
On the other hand, we easily see that
‖ TF (r) = Tf (r) +O
(
n+2∑
i=1
Tai(r)
)
= Tf (r) +O(log r).
Hence, it follows that
‖ Tf (r) ≤
(
n+2∑
i=1
n
mi
)
Tf (r) +O(log+ Tf (r)) +O(log r).
This implies that
‖ Tf (r) = O(log+ Tf (r)) +O(log r).
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1558 SI DUC QUANG, HA HUONG GIANG
Therefore
lim inf
r→+∞
Tf (r)/(log r) < +∞.
By again Lemma 3.1 we have the required extension of f.
The lemma is proved.
Proof of Theorem 1.1. Since {ai}n+2
i=1 are located in general position and f is linearly
nondegenerate over R{ai}, there exist nonzero meromorphic functions αi so that
n+2∑
i=1
αi(f, ai) = 0,
and ∑
i∈I
αi(f, ai) 6≡ 0 ∀I ( {1, . . . , n+ 2}.
We define the analytic subset S0 of D \ S by
S0 =
⋃
I({1,...,n+2}
{
z ∈ D \ S
∣∣∣∣∣ ∑
i∈I
αi(f(z), ai(z)) = 0
}
.
Then S0 is an analytic subset of codimension at least one of D \ S.
Put S̃ =
⋃
1≤i0<...<in≤n+2{z ∈ D |
⋂n
j=0 aij (z) 6= ∅}. It is easy to see that S̃ is an analytic
subset of codimension at least one of D. Denote by S1 the regular part of S
⋃
S̃ and S2 the singular
part of S
⋃
S̃. By [5], Corollary 3.3.44, it is enough to prove that f extends to a meromorphic
mapping on D \ S2.
Since the extendibility of the map f is a local property, it is suffices to prove that f is extendable
on a neighborhood of each point in S1.
For z0 ∈ S1, we take a small neighborhood U of z0 in D \ S2 so that U is biholomorphic with
∆×∆m−1. Then for convenience, we may assume that U = ∆×∆m−1 and S1∩U = {∞}×∆m−1.
Take a homogeneous coordinates (ω0 : . . . : ωn) of Pn(C) and set fi = ωi ◦f. It suffices to show
that for each 1 ≤ i ≤ n, fi
f0
extends meromorphically over ∆×∆m−1.
We easily see that there exists (a, b) ∈ C × Cm−1, a 6= 0, so that the complex line L =
= {(ta, z0 + tb)} satisfying L ∩ (U \ S) 6⊂ S0. Therefore, by changing the complex coordinates, we
may assume that ∆∗ × {z0} 6⊂ S0. It follows that there exists a neighborhood U1 of z0 in ∆m−1
so that ∆∗ × {z} 6⊂ S0 for all z ∈ U1. By choosing a smaller neighborhood if necessary, we may
assume that U1 = U = ∆∗ ×∆m−1. Then ∆∗ × {z} 6⊂ S3 for all z ∈ ∆m−1.
We consider the holomorphic curve f(·, z0) : z ∈ ∆∗ 7−→ f(z, z0), which intersect ai |∆∗×{z0}
with multiplicity at least mi for all 0 ≤ i ≤ n + 2. Therefore, the curve f(·, z0) and the family
{ai |∆∗×{z0}}ai∈A satisfy the assumption of Lemma 3.3.
By Lemma 3.3, f(·, z0) is extendable over ∆∗, hence
fi
f0
(·, z0) extends to a meromorphic function
on ∆ denoted again by
fi
f0
(·, z0). We put
fi
f0
(z1, z0) = z
µ(z0)
1 g(z1, z0), where µ(z0) ∈ Z and g(·, z0)
is a holomorphic function on ∆, g(∞, z0) 6= 0,∞. Take a small neighborhood U2 of z0. Then µ(z′)
is bounded in z′ ∈ U2. Then there is a neighborhood of (∞, z0), we may assume again that it is
∆×∆m−1, so that
fi
f0
is written as
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BIG PICARD THEOREM FOR MEROMORPHIC MAPPINGS WITH MOVING HYPERPLANES IN Pn(C) 1559
fi
f0
(z1, z
′) = (za1)g(z1, z
′),
where a ∈ Z, g is a nowhere vanishing holomorphic function on ∆∗ × ∆. g(∞, z′) 6= 0,∞ and
g(z1, z′) is holomorphic in z1 ∈ ∆∗ for each z′ ∈ ∆m−1.
We consider the expansion of g(z1, z′) in z1 at the point∞ as follows:
g(z1, z′) =
∞∑
i=0
bi(z
′)
(
1
z1
)i
.
Since g =
fi
za1f0
, which is a holomorphic function on ∆∗×∆m−1, it easy to see that each coefficient
bi(z
′) is holomorphic on ∆m−1. Hence g is holomorphic on ∆×∆m−1. Therefore
fi
f0
is meromorphic
on ∆×∆m−1.
The theorem is proved.
4. Big Picard theorem for the case of degenerate meromorphic mappings. In this section we
consider holomorphic mappings without the condition on linearly nondegeneracy of mappings, we
will prove the following theorem.
Theorem 4.1. Let f be a holomorphic mapping of a domain D \ S into Pn(C), where D is a
domain in Cm and S is an analytic subset of codimension one of D with only normal crossings. Let
N be a positive integer. Let A = {a0, . . . , aq−1} be a set of q, q ≥ 2N + 1, moving hyperplanes on
D of Pn(C) located in pointwise N -subgeneral position with respect to f. Assume that f intersects
each ai on D \ S with multiplicity at least mi, where m0, . . . ,mq−1 are fixed positive integers and
may be +∞, with
q−1∑
i=0
1
mi
<
q − 2N − 1
Lf
+ 1.
Then f extends to a holomorphic mapping f̃ from D into Pn(C).
In order to prove Theorem 4.1, we need some following.
Definition 4.1 ([11], Definition 3.1). Let Ω be a hyperbolic domain and let M be a complete
complex Hermitian manifold with metric ds2
M . A holomorphic mapping f(z) from Ω into M is said
to be a normal holomorphic mapping from Ω into M if and only if there exists a positive constant C
such that for all z ∈ Ω and all ξ ∈ Tz(Ω),
ds2
M
(
f(z), df(z)(ξ)
)
≤ CKΩ(z, ξ),
where df(z) is the mapping from Tz(Ω) into Tf(z)(M) induced by f and KΩ denotes the infinitesimal
Kobayashi metric on Ω.
Lemma 4.1 (see [11]). Let f be a holomorphic mapping from a bounded domain Ω in Cm into
Pn(C) such that for every sequence of holomorphic mappings ϕk(z) from the unit disc U in C into
Ω, the sequence {f ◦ ϕk(z)}∞k=1 from U into Pn(C) is a normal family on U. Then f is a normal
holomorphic mapping from Ω into Pn(C).
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1560 SI DUC QUANG, HA HUONG GIANG
Theorem 4.2 ([1], Theorem 3.1, [9], Theorem 2.5). Let Ω be a domain in Cm. Let M be a
compact complex Hermitian space. Let F ∈ Hol(Ω,M). Then the family F is not normal if and only
if there exist sequences {pj} ∈ Ω with {pj} → p0, (fj) ⊂ F , {ρj} ⊂ R with ρj > 0 and {ρj} → 0
such that
gj(ξ) := fj(pj + ρjξ)
converges uniformly on compact subsets of Cm to a nonconstant holomorphic map g : Cm →M.
The following theorem is due to Noguchi [6].
Theorem 4.3 ([6], Theorem 3.1). Let f be a linearly nondegenerate holomorphic mapping of
Cm into Pk(C) and let {H0, . . . Hq−1} be q, q ≥ 2N − k + 1, hyperplanes of Pk(C) located in
N -subgeneral position with respect to f. Then the following holds:
‖ (q − 2N + k − 1)Tf (r) ≤
q−1∑
i=0
N (k)(r, div(f,Hi)) + o(Tf (r)).
For our purpose, we need a reformulation of Theorem 4.3 as follows.
Lemma 4.2. Let f be a holomorphic mapping of Cm into Pn(C) and let {H0, . . . Hq−1} be q,
q ≥ 2N −Lf + 1, hyperplanes of Pn(C) located in N−subgeneral position with respect to f. Then
the following holds:
‖ (q − 2N + Lf − 1)Tf (r) ≤
q−1∑
i=0
(f,Hi) 6≡0
N (Lf )(r, div(f,Hi)) + o(Tf (r)).
Proof. We give a sketch of its proof as follows. Denote by Df the linearly span of f(C). Then
we may consider Df as a complex projective space of dimension Lf . Set Q = {j; (f,Hj) ≡ 0}. It
is clear that an index j ∈ Q if and only if Df ⊂ Hj . For each i 6∈ Q, we set H∗i = Hi ∩Df , which
is a hyperplane in Df , and easily see that
div(f,Hi) = div(f,H∗i ),
here for the right-hand side of the equality we consider f as a map of C into Df . We may verify
that {H∗i ; 0 ≤ i ≤ q − 1, i 6∈ Q} is located in (N − ]Q)-subgeneral position in Df . Indeed, for any
subset {i0, . . . , iN−]Q} of {0, . . . , q − 1}\Q, we have
N−]Q⋂
j=0
H∗ij = Df ∩
N−]Q⋂
j=0
Hij = Df ∩
⋂
i∈Q
Hi ∩
N−]Q⋂
j=0
Hij = ∅.
Applying Theorem 4.3, we obtain
‖ ((q − ]Q)− 2(N − ]Q) + Lf − 1)Tf (r) ≤
q−1∑
i=0
i 6∈Q
N (Lf )(r, div(f,H∗i )) + o(Tf (r)).
This easily implies that
‖ (q − 2N + Lf − 1)Tf (r) ≤
q−1∑
i=0
i 6∈Q
N (Lf )(r, div(f,Hi)) + o(Tf (r)).
The lemma is proved.
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BIG PICARD THEOREM FOR MEROMORPHIC MAPPINGS WITH MOVING HYPERPLANES IN Pn(C) 1561
Proof of Theorem 4.1. For z0 ∈ S, we take a relative compact subdomain Ω containing z0 of
D. It suffices to prove that f extends over Ω \ S to a holomorphic mapping.
Firstly, we shall prove that f is normal on Ω \ S. Indeed, suppose that f is not normal on Ω \ S,
then there exists a sequence of holomorphic mappings {ϕi : U → Ω\S}∞j=1 such that {f ◦ϕj} is not
normal, where U denotes the unit disc in C. By Lemma 4.2, we may assume that there exist sequences
{pj} ∈ U, {rj} ∈ R with rj > 0 and rj ↘ 0, pj → p0 ∈ U such that gj(ξ) := f ◦ ϕj(pj + rjξ)
converges uniformly on compact subsets of C to a nonconstant holomorphic mapping g of C into
Pn(C). Because Ω \ S is bounded, then {ϕj} is a normal family of holomorphic mappings. Hence,
there exists a subsequence (again denoted by {ϕj}) of {ϕj} which converges uniformly on compact
subsets of U to a holomorphic ϕ : U → Ω. Then limj→∞ ϕj(pj + rjξ) = ϕ(p0) ∈ Ω. Since f(z)
intersects ai(z) with multiplicity at leastmi, gj(ξ) intersects ai(ϕj(pj+rjξ)) with multiplicity at least
mj for all 0 ≤ i ≤ q − 1 and 1 ≤ j. By Hurwitz’s theorem g intersects ai(ϕ(p0)) with multiplicity
at least mj or g(C) is included in ai(ϕ(p0)) for all 0 ≤ i ≤ q − 1.
Applying Lemma 4.2, we obtain
‖ (q − 2N + Lg − 1)Tg(r) ≤
∑
(g,aj(ϕ(p0))) 6≡0
N (Lg)(r, div(g,Hj)) + o(Tg(r)) ≤
≤
∑
(g,aj(ϕ(p0))) 6≡0
Lg
mj
N(r, div(g,Hj)) + o(Tg(r)) ≤
≤
∑
(g,aj(ϕ(p0))) 6≡0
Lg
mj
Tg(r) + o(Tg(r)).
Letting r −→ +∞, we get
q − 2N + Lg − 1 ≤
q−1∑
j=0
Lg
mj
⇔
q−1∑
j=0
1
mj
≥ q − 2N − 1
Lg
+ 1.
It is clear that Lg ≤ Lf , then
∑q−1
j=0
1
mj
≥ q − 2N − 1
Lf
+ 1. This is a contradiction. Hence f is
normal on Ω \ S.
By the assumption of Theorem 4.1, S ∩ Ω is an analytic subset of domain Ω with codimension
1, whose singularities are normal crossings. Then f extends to a holomorphic mapping from Ω into
Pn(C) by Theorem 2.3 in Joseph and Kwack [4].
Theorem 4.1 is proved.
Acknowledgements. The authors would like to thank Professor Do Duc Thai for this valuable
suggestions.
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Received 29.10.12
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