Extension of holomorphic mappings for few moving hypersurfaces
We prove the big Picard theorem for holomorphic curves from a punctured disc into $P^n(C)$ with $n + 2$ hypersurfaces. We also prove a theorem on the extension of holomorphic mappings in several complex variables into a submanifold of$P^n(C)$ with several moving hypersurfaces.
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| Дата: | 2012 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2012
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2584 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508505877774336 |
|---|---|
| author | Si, Duc Quang Сі, Дук Куанг |
| author_facet | Si, Duc Quang Сі, Дук Куанг |
| author_sort | Si, Duc Quang |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:30:02Z |
| description | We prove the big Picard theorem for holomorphic curves from a punctured disc into $P^n(C)$ with $n + 2$ hypersurfaces.
We also prove a theorem on the extension of holomorphic mappings in several complex variables into a submanifold of$P^n(C)$ with several moving hypersurfaces. |
| first_indexed | 2026-03-24T02:26:17Z |
| format | Article |
| fulltext |
UDC 517.5
Si Duc Quang (Hanoi Univ. Education, Vietnam)
EXTENSION OF HOLOMORPHIC MAPPINGS
FOR FEW MOVING HYPERSURFACES*
ПРОДОВЖЕННЯ ГОЛОМОРФНИХ ВIДОБРАЖЕНЬ
ДЛЯ ДЕКIЛЬКОХ ГIПЕРПОВЕРХОНЬ, ЩО РУХАЮТЬСЯ
We prove the big Picard theorem for holomorphic curves from a punctured disc into Pn(C) with n + 2 hypersurfaces.
We also prove a theorem on the extension of holomorphic mappings in several complex variables into a submanifold of
Pn(C) with several moving hypersurfaces.
Доведено велику теорему Пiкара для голоморфних кривих iз проколотого круга в Pn(C) iз n + 2 гiперповерх-
нями. Також доведено теорему про продовження голоморфних вiдображень вiд декiлькох комплексних змiнних у
пiдбагатовид Pn(C) з декiлькома гiперповерхнями, що рухаються.
1. Introduction. Picard proved the following theorems for meromorphic functions in one complex
variable.
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] gave a Big Picard’s theorem for holomorphic
mappings from a complex manifold into Pn(C) as follows.
Theorem C (Theorem A [3]). Let M 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.
By using a criterion on normality and by applying little Picard theorems for holomorphic map-
pings, Z. H. Tu generalized the above theorems to the case of moving hyperplanes as follows.
Theorem D (Theorem 2.2 [11]). 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 f extends to a holomorphic mapping from D into Pn(C).
*This work was supported by a NAFOSTED grant of Vietnam.
c© SI DUC QUANG, 2012
392 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
EXTENSION OF HOLOMORPHIC MAPPINGS FOR FEW MOVING HYPERSURFACES 393
We would like to note that in Theorem D, the number of hyperplanes is assumed to be at least
2n+1 and this assumption plays a very essential role in the proof. Then, the following question arises
naturally: “Are there any Big Picard’s theorems which are analogous to Theorem C or Theorem D in
the case where the moving hyperplanes are replaced by moving hypersurfaces and the number q is
replaced by a smaller one ?”
In the present paper we will give some positive answers for this question. First of all, let us recall
some following.
Denote by HD the ring of all holomorphic functions on a domain D in Cm. Let Q be a homoge-
neous polynomial in HD[x0, . . . , xn] of degree d ≥ 1. Denote by Q(z) the homogeneous polynomial
over C obtained by substituting a specific point z ∈ D into the coefficients of Q. We also call a
moving hypersurface in Pn(C) on D each homogeneous polynomial Q ∈ HD[x0, . . . , xn] such that
the coefficients of Q have no common zero point.
Let Q1, . . . , Qq be q moving hypersurfaces of Pn(C) on D.
Set
Td :=
{
(i0, . . . , in) ∈ Nn+1 |i0 + i1 + . . .+ in = d
}
.
Assume that
Qj(z) =
∑
I∈Tdj
ajI(z)x
I ,
where ajI are holomorphic functions on D without common zeros, xI = xi00 . . . x
in
n for x =
= (x0, . . . , xn) and I = (i0, . . . , in) ∈ Tdj , dj = deg(Qj).
Denote by R{Qj} the smallest field which contains C and all functions
ajI
ajJ
with aiJ 6≡ 0.
Sometime we write R for R{Qj} if there is no confusion.
We say that moving hypersurfaces {Qj}qj=1 in Pn(C) are located in general position (resp. in
pointwise general position) on a subset Ω ⊂ D if there exists z ∈ Ω (resp. for all z ∈ Ω) such that
for any 1 ≤ j0 < . . . < jn ≤ q the system of equations
Qji(z)(w0, . . . , wn) = 0, 0 ≤ i ≤ n,
has only the trivial solution w = (0, . . . , 0) in Cn+1.
Let f be a meromorphic mapping of D into Pn(C) and let Q be a moving hypersurface of Pn(C)
on D defined by Q(z) =
∑
I∈Td
ajI(z)x
I , where d is the degree of homogeneous polynomial Q.
For z0 ∈ D, take a reduced representation f = (f0 : . . . : fn) of f on a neighborhood Uz0 of z0 and
set Q(f)(z) = Q(z)(f0(z), . . . , fn(z)) on Uz0 . We define divQ(f)(z) = div(Q(f0, . . . , fn))(z) if
Q(f) 6≡ 0 and divQ(f)(z) = ∞ if Q(f) ≡ 0. Thus, divQ(f) is well-defined on D independently
of the choice of reduced representations of f. If divQ(f)(z) ≥ mj for all z ∈ D, we say that f
intersects Q on D with multiplicity at least mj .
We set punctured discs on Ĉ = C ∪ {∞} around∞ by
∆∗ = {z ∈ C : |z| ≥ 1},
∆∗(t) = {z ∈ C : |z| ≥ t}, 1 ≤ t ≤ ∞,
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394 SI DUC QUANG
and set ∆ = ∆∗ ∪ {∞}. For a moving hypersurface Q in Pn(C) on ∆∗ defined by Q(z) =
=
∑
I∈Td
ajI(z)x
I , we say that Q is a moving hypersurface in Pn(C) on ∆ if all coefficients ajI
are extendable over ∆.
Our first aim of this paper is to show a Big Picard’s theorem for holomorphic curve from a
punctured disc with only n+ 2 hypersurfaces. Namely, we will prove the following theorem.
Theorem 1. Let f be a holomorphic curve from the punctured disc ∆∗ into Pn(C), and let
Q1, . . . , Qn+2 be n + 2 hypersurfaces in Pn(C) on ∆ located in general position such that f is
algebraically nondegenerate over R{Qi}. Assume that f intersects each Qi on ∆∗ with multiplicity
at least mi, where m1, . . . ,mn+2 are fixed positive integers and may be +∞, with
n∑
i=1
1
mi
<
1
M
,
where M = (nd+[(n+1)2(2n−1)(dε)−1]d)n. Then f extends at∞ to a holomorphic curve f̃ from
∆ = ∆∗ ∪ {∞} to Pn(C).
In the case of moving hypersurfaces and an arbitrary meromorphic mapping from a domain in
Cm into a subvariety V of Pn(C), we shall prove the following, which is a generalization of the
above result of H. Fujimoto.
Theorem 2. Let f be a holomorphic mapping of a domain D \ S into X, where D is a
domain in Cm, S is an analytic subset of co-dimension one of D, whose singularities are only
normal crossings, and X is an irreducible subvariety of Pn(C). Let Q0, . . . , Qq−1 be q moving
hypersurfaces of Pn(C) on D located in pointwise subgeneral position with respect to X. Assume
that f does not intersect each Qi on D \S for all 1 ≤ i ≤ q− 1. If q ≥ 2 dimX+ 1. Then f extends
to a holomorphic mapping f̃ from D into Pn(C).
2. Notations. (a) We set punctured discs on Ĉ = C ∪ {∞} around∞ by
∆∗ = {z ∈ C : |z| ≥ 1},
∆∗(t) = {z ∈ C : |z| ≥ t}, t ≥ 1,
and set
Γ(r) = {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 [5] (§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.
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EXTENSION OF HOLOMORPHIC MAPPINGS FOR FEW MOVING HYPERSURFACES 395
(b) 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 E by Supp (E) = ∪ν 6=0pν . Let
k be a positive integer or +∞. We define the divisor E(k) by
E(k) :=
∑
min{µν , k}pν ,
and define 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).
(c) Let f : ∆∗ → Pn(C) be a holomorphic curve. For an arbitrary fixed homogeneous coordinates
(w0 : . . . : wn) of Pn(C), it is easy to see that 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 Fubibi – Study form of Pn(C). The order function or characteristic function of
f with respect to Ω is defined by
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 the following:
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 Q be a hypersurface in Pn(C) given by Q(x) =
∑
I∈Td
aIx
I , where the constants aI are
not all zeros and d is the degree of Q. We set Q(f) =
∑n
i∈Td
aIf
I , where f I = f i00 . . . f inn for
I = (i0, . . . , in) ∈ Td. Assume that Q(f) 6≡ 0, we define the proximity function of f with respect to
Q by
mf (r,Q) =
1
2π
∫
Γ(r)
log
‖f‖d
|Q(f)|
dθ − 1
2π
∫
Γ(1)
log
‖f‖d
|Q(f)|
dθ.
Applying (2.1) to ξ = log |Q(f)|, we have
N(r, divQ(f)) =
1
2π
∫
Γ(r)
log |Q(f)|dθ − 1
2π
∫
Γ(1)
log |Q(f)|θ − (log r)
∫
Γ(1)
dc log |Q(f)|. (2.4)
Combining (2.2) and (2.4), we have the First Main Theorem as follows:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
396 SI DUC QUANG
dTf (r) = N(r, divQ(f)) +mf (r,Q) + (log r)
∫
Γ(1)
dc log
(
‖f‖d
|Q(f)|
)
. (2.5)
(d) 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 C into P1(C), there is a fact that
Tϕ(r) = T (r, ϕ) + O(log r).
Theorem 3 (lemma on logarithmic derivative [5]). 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. Second main theorem for holomorphic curves from a punctured disc. Firstly, we prove a
Second Main Theorem for holomorphic curves from the punctured disc ∆∗ into Pn(C) for hyper-
surfaces with truncated multiplicities as follows.
Theorem 4. Let f be an algebraically nondegenerate holomorphic curve from the punctured
disc ∆∗ into Pn(C) and let Qi, 1 ≤ i ≤ q, be q hypersurfaces of Pn(C) of degree di on ∆∗ located
in general position, q ≥ n+ 2. Then for every ε > 0, the following holds
‖ (q − n− 1− ε)Tf (r) ≤
q∑
i=1
1
di
N (M0) (r, div(Qi(f))) +O(log r),
where M0 = (nd+ [(n+ 1)2(2n − 1)(dε)−1]d)n with d is the least common multiple of the d′is.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
EXTENSION OF HOLOMORPHIC MAPPINGS FOR FEW MOVING HYPERSURFACES 397
Proof. Take (ω0 : . . . : ωn) be a homogeneous coordinates of Pn(C) and take (f0 : . . . : fn) be
a reduced representation of f on a neighborhood of ∆∗ in Cm. Replacing Qj by Q
d/dj
j if necessary,
we may assume that Q1, . . . , Qq have the same degree of d.
Given z ∈ Cm, there exists a renumbering {i1, . . . , iq} of the indices {1, . . . , q} such that
|Qi1(f)(z)| ≤ |Qi2(f)(z)| ≤ . . . ≤ |Qiq(f)(z)|.
We denote γ := (Qi1 , . . . , Qin). Since {Qj}qj=1 are in general position, by Hilbert’s Nullstellensatz
that for any integer k, 0 ≤ k ≤ n, there is an integer mk ≥ d such that
Xmk
k =
n+1∑
j=1
bjkQij (X0, . . . , Xn),
where bjk, 1 ≤ j ≤ n + 1, 0 ≤ k ≤ n, are homogeneous forms with coefficients in C of degree
mk − d. So
|fk(z)|mk ≤ c1‖f(z)‖mk−d max{|Qi1(f)(z)|, . . . , |Qin+1(f)(z)|} =
= c1‖f(z)‖mk−d|Qin+1(f)(z)|,
where c1 is a positive constant which depends only on the coefficients of Qi, 1 ≤ i ≤ q. Therefore
‖f(z)‖d ≤ c1|Qin+1(f)(z)|. (3.1)
Fix big integer N, which will be chosen later, such that N divisible by d, denote by VN the space of
homogeneous polynomials in C[X0, . . . , Xn] of degree N. Arrange, by the lexicographic order, the
n-tuples (j) = (j1, . . . , jn) of nonnegative integers such that σ(j) :=
∑n
k=1
jk ≤
N
d
. Define the
spaces
W(j) =
∑
(e)≥(j)
Qj1i1 . . . Q
jn
in
VN−dσ(e).
We put ∆(j) := dim
W(j)
W(j′)
, where (j′) follows (j) in the ordering. From Lemma 3 [2], we have
∆(j) = dn, (3.2)
provided dσ(j) < N − nd.
Set M := dimVN . We now chose a suitable basis as follows: We start with the last nonzero W γ
(j),
pick any basic of it. Then we continue inductively as follows, for (j′) > (j) such that dσ(j), dσ(j′) ≤
≤ N. Assume that we have chosen a basic of W γ
(j′), we pick representatives in W γ
(j) of the basic
of W γ
(j)/W
γ
(j′) which are the form Qj1i1 . . . Q
jn
in
q, where q ∈ VN−dσ(j). We extend the previously
constructed basic in W γ
(j′) by adding these representations, then we have a basic of W γ
(j). If W γ
(j) =
= VN then we stop the process and we obtain a basic of VN .
Now we estimate log
∏M
t=1
|ψγt (f)(z)|. With ψ be an element of the basic constructed with
respect to W γ
(j)/W
γ
(j′), ψ = Qj1i1 . . . Q
jn
in
q, q ∈ VN−dσ(j). Then we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
398 SI DUC QUANG
|ψ(f)(z)| ≤ C2‖f(z)‖N−dσ(j)|Qi1(f)(z)|j1 . . . |Qin(f)(z)|jn ,
then
M∏
t=1
|ψγt (f)(z)| ≤ C3‖f(z)‖
∑
(j) ∆γ
(j)
(N−dσ(j)
n∏
k=1
|Qik(f)(z)|
∑
(j) ∆γ
(j)
jk , (3.3)
where C2, C3 are constants which depend only on N and the coefficients of {Qi}qi=1
(
the sum and
product are taken over all n-tuples (i), such that σ(i) ≤ N
d
)
.
We fix φ1, . . . , φM , a basic of VN , ψ
γ
t (f) = Lγt (F ), where Lγt are linear forms and
F = (φ1(f) : . . . : φM (f)).
We set
bγk =
∑
(j)
∆γ
(j)jk, 1 ≤ j ≤ n,
aγ =
∑
(j)
∆γ
(j)(N − dσ(j)),
where the sums are taken over all n-tuples (j) such that σ(j) ≤ N/d. We note that
aγ +
n∑
k=1
dbγk = NM.
From (3.3) we have that
log
M∏
t=1
|Lγt (F )(z)| ≤ log
n∏
j=1
|Qij (f)(z)|b
γ
j
+ log ‖f(z)‖aγ + C4,
where C4 is a constant which depends only on N and the coefficients of {Qi}qi=1.
We set b = mink,γ b
γ
k . Because f is algebraically non degenerate over C,
F = (φ1(f) : . . . : φM (f))
is linearly non degenerate over C, then we have
W (φi(f)) = det
(
∂i(φj(f))
∂zi
)
1≤i,j≤M
6≡ 0.
We also have
log
‖f(z)‖(q−n)db|W (φi(f))(z)|
(
∏q
j=1 |Qj(f)(z)|b)‖f(z)‖(NM−ndb)
≤ log
W (φi(f))(z)|
(
∏n
k=1 |Qik(f)(z)|b)‖f(z)‖(NM−ndb)
≤
≤ log
W (φi(f))(z)|C5
(
∏n
k=1 |Qik(f)(z)|b
γ
k )‖f(z)‖(NM−d
∑n
k=1 b
γ
k)
≤
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
EXTENSION OF HOLOMORPHIC MAPPINGS FOR FEW MOVING HYPERSURFACES 399
≤ log
W (φi(f))(z)|C3C5∏M
i=1 |ψ
γ
i (f)(z)|
≤ log
W (ψγi (f))(z)|C6∏M
i=1 |ψ
γ
i (f)(z)|
, (3.4)
where C5, C6 are constants, which are depend only on N and {Qi}qi=1.
From (3.4), for all z ∈ C \ I(f), which are not zero of Qi(f), 1 ≤ i ≤ q, we have
log
‖f(z)‖(q−n)db|Wα(φγi (f))(z)|(∏q
j=1
|Qj(f)(z)|b
)
‖f(z)‖(NM−ndb)
≤
∑
γ
log+
Wα(ψγi (f))(z)|C6∏M
i=1
|ψγi (f)(z)|
.
Integrating both sides of the above inequality over Γ(r), we obtain∥∥∥∥(q − NM
db
)
Tf (r) ≤
≤
q∑
j=1
1
d
N(r, divQj(f))− 1
db
N(r, div(Wα(φi(f)))) +O(log+(Tf (r))) + C log r, (3.5)
where C is a positive constant (may be depend on f and Qi).
We now have some estimates. First,
M =
(N+n
n
)
=
(N + 1) . . . (N + n)
1 . . . n
.
Second, since the number of nonnegative integer p-tuples with summation ≤ T is equal to the number
of nonnegative integer (p+1)-tuples with summation exactly equal T ∈ Z, which is
(
T +m
m
)
, since
the sum below is independent of k, we have that
bγk =
∑
σ(j)leN/d
∆(j)jk ≥
∑
σ(j)≤N/d−n
∆(j)jk =
=
∑
σ(j)≤N/d−n
dnjk =
dn
n+ 1
∑
σ(j)≤N/d−n
n+1∑
k=1
jk =
=
dn
n+ 1
∑
σ(j)≤N/d−n
N
d
=
dnN
(n+ 1)d
(N/d
n
)
=
dnN(N/d− 1) . . . (N/d− n)
1 . . . (n+ 1)d
.
This implies that
NM
db
≤ (n+ 1)
(N + 1) · · · (N + n)
(N − d) · · · (N − nd)
≤
≤ (n+ 1)
n∏
k=1
n+ k
N − (n+ 1)d+ kd
≤ (n+ 1)
(
N + 1
N − nd
)n
.
We chose
N ≥ nd+
[
n+ 1/d
(1 + ε/(n+ 1))1/n − 1
]
d.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
400 SI DUC QUANG
Then N divisible by d and one gets
N ≥ nd+
[
(n+ 1)2(2n − 1)(dε)−1
]
d and
(
q − NM
db
)
≥ (q − n− 1− ε).
Thus, from (3.5) we obtain
‖(q − n− 1− ε)Tf (r) ≤
≤
q∑
j=1
1
d
N(r, divQj(f))− 1
db
N(r, div(Wα(φi(f)))) +O(log+(Tf (r))). (3.6)
We now estimate
(∑q
j=1
div(Qj(f))− 1
b
div(W (φi(f)))
)
. Fix z ∈ Cm, we may assume that
div(Qj1(f))(z) ≥ . . . ≥ div(Qjk(f))(z) > 0 = div(Qjk+1
(f))(z) = . . . = div(Qjq(f))(z),
where 0 ≤ k ≤ n (k may be zero). Put γ = (Qj1 , . . . , Qjn), then we have
div(W (φi(f)))(z) = div(W (ψγi (f)))(z) ≥
M∑
t=1
max{div(ψγt (f))(z)−M, 0}.
For ψ = Qi1j1 . . . Q
in
jn
q ∈ {ψγt }Mt=1, we have
ψ(f)(z) = Qi1j1(f)(z) . . . Qinjn(f)(z).q(f)(z).
Hence
max{div(ψ(f))(z)−M, 0} ≥
n∑
k=1
max{div(Qikjk(f))(z)−M, 0} ≥
≥
n∑
k=1
ik max{div(Qjk(f))(z)−M, 0}.
This implies that
M∑
t=1
max{div(ψt(f))(z)−M, 0} ≥
∑
(i)
∆γ
(i)
n∑
k=1
ik max{div(Qjk(f))(z)−M, 0} =
=
n∑
k=1
bγk max{div(Qjk(f))(z)−M, 0} ≥
≥
n∑
k=1
bmax{div(Qjk(f))(z)−M, 0}.
Hence
q∑
j=1
divQj(f)(z)− 1
b
divWα(φj(f))(z) ≤
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
EXTENSION OF HOLOMORPHIC MAPPINGS FOR FEW MOVING HYPERSURFACES 401
≤
q∑
j=1
(divQj(f)(z)−max{divQj(f)(z)−M, 0}) =
=
q∑
j=1
divQj(f)[M ](z). (3.7)
From (3.7) we obtain
q∑
j=1
N(r, divQj(f))− 1
b
N(r, div(Wα(φj(f)))) ≤
q∑
j=1
N (M)(r,Qj(f)). (3.8)
Combining (3.6) and (3.8), we have∥∥∥∥(q − n− 1− ε)Tf (r) ≤
q∑
j=1
1
d
N (M)(r,Qj(f)) +O(log+(Tf (r))).
One can be estimated that
M ≤
(
N + n
N
)
≤ Nn ≤
(
nd+ [(n+ 1)2(2n − 1)(dε)−1]d
)n ≤M0.
Theorem 4 is proved.
4. Proof of Theorem 1. Let f : ∆∗ → V be a holomorphic curve into a complex projective
algebraic variety V. We know the following characterization of a removable singularity (see [5]).
Lemma 1. Let f : ∆∗ → V be as above and let Tf (r) be a characteristic function with respect
an ample line bundle over V. Then f extends at ∞ to a holomorphic curve f̃ from ∆ = ∆∗ ∪ {∞}
into V if and only if
lim inf
r→∞
Tf (r)/(log r) <∞.
Proof of Theorem 1. For 0 < ε < 1 −
∑n+2
i=1
M
midi
, it follows from Theorem 4 and the
assumption that
‖(1− ε)Tf (r) ≤
n+2∑
j=1
1
di
N (M)(r, div(Qi(f))) +O(log Tf (r)) +O(log r) ≤
≤
n+2∑
j=1
M
midi
N(r, divQi(f)) +O(log Tf (r)) +O(log r) ≤
≤
n+2∑
j=1
M
midi
Tf (r) +O(log Tf (r)) +O(log r).
This implies that
‖Tf (r) = O(log Tf (r)) +O(log r).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
402 SI DUC QUANG
Therefore
lim inf
r→+∞
Tf (r)/(log r) < +∞.
By Lemma 1 we have the required extension of f.
Theorem 1 is proved.
5. Proof of Theorem 2. In order to prove Theorem 2, we need some following.
Definition 1 (Definition 3.1 [11]). 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 2 (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).
Theorem 5 (Theorem 3.1 [1], Theorem 2.5 [10]). Let Ω be a domain in Cm. Let M be a com-
pact 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 non-constant holomorphic map g : Cm →M.
Proof of Theorem 2. 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 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 non-constant holomorphic mapping g of C into
Pn(C). Because Ω \ S is bounded, {ϕj} is a normal family of holomorphic mappings. Hence, there
exists a sub-sequence (again denoted by {ϕj}) of {ϕj} which converges uniformly on compact
subsets of U to a holomorphic mapping ϕ : U → Ω. Then limj→∞ ϕj(pj + rjξ) = ϕ(p0) ∈ Ω.
Since f(z) does not intersect Qi(z), then g does not intersect Qi(ϕ(p0)) or g(C) is included in
Qi(ϕ(p0)) for all 0 ≤ i ≤ q − 1 by Hurwitz’s theorem. Hence, there exists a subset I of {1, . . . , q}
such that g(C) ⊂ (∩i∈IQi(ϕ(p0)) \ ∪i 6∈IQi(ϕ(p0))) ∩X. By Corollary 1.4 [7], we have that the set
∩i∈IQi(ϕ(p0)) \ ∪i 6∈IQi(ϕ(p0)) is hyperbolic imbedded into X. Then g must be constant. This is a
contradiction. Hence, f is normal.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
EXTENSION OF HOLOMORPHIC MAPPINGS FOR FEW MOVING HYPERSURFACES 403
By the assumption of Theorem 2, S ∩ Ω is an analytic subset of domain Ω with codimension
one, 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 2 is proved.
Remark. Let f be a holomorphic mapping of a domain D \ S into X, where D is a domain in
Cm, S is an analytic subset of co-dimension at least two of D and X is an irreducible subvariety of
Pn(C). Let Q be a moving hypersurface of Pn(C) on D. Assume that f does not intersect Q on D,
then f extends to a holomorphic mapping of D into X.
Indeed, by Corollary 3.3.44 [6], f extends to a meromorphic mapping of D into X (denoted
again by f ). It suffices to show that f is holomorphic on D.
Suppose that f is not holomorphic on D. We denote by I the indeterminancy locus of f which
is a non empty analytic subset of codimension two of D.
It is easy to see that I ⊂ Supp (divQ(f)). Then Supp (divQ(f)) is a non empty analytic
subset of codimension one of D. Therefore Supp (divQ(f)) ∩ (D \ S) 6= ∅. This contradicts to the
assumption that f does not intersect Q on D \ S. Hence f is holomorphic on D.
Acknowledgements. The author would like to thank Professors Junjiro Noguchi and Do Duc
Thai for their valuable advice and suggestions concerning this material.
1. Aladro G., Krantz S. G. A criterion for normality in Cn // J. Math. Anal. and Appl. – 1991. – 161. – P. 1 – 8.
2. An T. T. H., Phuong H. T. An explicit estimate on multiplicity truncation in the second main theorem for holomorphic
curves encountering hypersurfaces in general position in projective space // Houston J. Math. – 2009. – 35. –
P. 775 – 786.
3. Fujimoto H. Extensions of the big Picard’s theorem // Tohoku Math. J. – 1972. – 24. – P. 415 – 422.
4. Joseph J., Kwack M. H. Extension and convergence theorems for families of normal maps in several complex
variables // Proc. Amer. Math. Soc. – 1997. – 125. – P. 1675 – 1684.
5. Noguchi J. Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties // Nagoya Math. J. –
1981. – 83. – P. 213 – 233.
6. Noguchi J., Ochiai T. Introduction to geometric function theory in several complex variables // Trans. Math. Monogr.
– Providence, Rhode Island: Amer. Math. Soc., 1990. – 80.
7. Noguchi J., Winkelmann J. Holomorphic curves and integral points off divisors // Math. Z. – 2002. – 239. –
P. 593 – 610.
8. Ru M. A defect relation for holomorphic curves intersecting hypersurfaces // Amer. J. Math. – 2004. – 126. –
P. 215 – 226.
9. Stoll W. Normal families of non-negative divisors // Math. Z. – 1964. – 84. – P. 154 – 218.
10. Thai D. D., Trang P. N. T., Huong P. D. Families of normal maps in several complex variables and hyperbolicity of
complex spaces // Complex Variables and Elliptic Equat. – 2003. – 48. – P. 469 – 482.
11. Tu Z. H., Li P. Big Picard Theorems for holomorphic mappings of several complex variables into PN (C) with
moving hyperplanes // J. Math. Anal. and Appl. – 2006. – 324. – P. 629 – 638.
Received 27.09.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
|
| id | umjimathkievua-article-2584 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:17Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7a/6a8e38dcc3e90b8e90ec380e8a97197a.pdf |
| spelling | umjimathkievua-article-25842020-03-18T19:30:02Z Extension of holomorphic mappings for few moving hypersurfaces Продовження голоморфних вiдображень для декiлькох гiперповерхонь, що рухаються Si, Duc Quang Сі, Дук Куанг We prove the big Picard theorem for holomorphic curves from a punctured disc into $P^n(C)$ with $n + 2$ hypersurfaces. We also prove a theorem on the extension of holomorphic mappings in several complex variables into a submanifold of$P^n(C)$ with several moving hypersurfaces. Доведено велику теорему Пiкара для голоморфних кривих iз проколотого круга в $P^n(C)$ iз $n + 2$ гiперповерхнями. Також доведено теорему про продовження голоморфних вiдображень вiд декiлькох комплексних змiнних у пiдбагатовид $P^n(C)$) з декiлькома гiперповерхнями, що рухаються. Institute of Mathematics, NAS of Ukraine 2012-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2584 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 3 (2012); 392-403 Український математичний журнал; Том 64 № 3 (2012); 392-403 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2584/1927 https://umj.imath.kiev.ua/index.php/umj/article/view/2584/1928 Copyright (c) 2012 Si Duc Quang |
| spellingShingle | Si, Duc Quang Сі, Дук Куанг Extension of holomorphic mappings for few moving hypersurfaces |
| title | Extension of holomorphic mappings for few moving hypersurfaces |
| title_alt | Продовження голоморфних вiдображень для декiлькох гiперповерхонь, що рухаються |
| title_full | Extension of holomorphic mappings for few moving hypersurfaces |
| title_fullStr | Extension of holomorphic mappings for few moving hypersurfaces |
| title_full_unstemmed | Extension of holomorphic mappings for few moving hypersurfaces |
| title_short | Extension of holomorphic mappings for few moving hypersurfaces |
| title_sort | extension of holomorphic mappings for few moving hypersurfaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2584 |
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