Algebraic dependences of meromorphic mappings in several complex variables
We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces.
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| author | Pham, Duc Thoan Pham, Viet Duc Фам, Дік Тон Фам, В'єт Дік |
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| description | We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces. |
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UDC 517.946
Duc Thoan Pham (Hanoi Nat. Univ. Education, Vietnam),
Viet Duc Pham (Thai Nguyen Univ. Education, Vietnam)
ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS
IN SEVERAL COMPLEX VARIABLES*
АЛГЕБРАЇЧНА ЗАЛЕЖНIСТЬ МЕРОМОРФНИХ
ВIДОБРАЖЕНЬ ДЛЯ БАГАТЬОХ КОМПЛЕКСНИХ ЗМIННИХ
In this article, some algebraic dependence theorems of meromorphic mappings in several complex variables
into the complex projective spaces are given.
Наведено деякi теореми про алгебраїчну залежнiсть мероморфних вiдображень для багатьох комплекс-
них змiнних на комплекснi проективнi простори.
1. Introduction. The theory on algebraic dependences of meromorphic mappings in
several complex variables into the complex projective spaces for fixed targets is studied
by Wilhelm Stoll [1]. Later, Min Ru [2] generalized Stoll’s result to holomorphic curves
into the complex projective spaces for moving targets and show some unicity theorems
of holomorphic curves into the complex projective spaces for moving targets. As far as
we know, they are the first results on the unicity problem for moving targets. We now
state his remarkable results.
Let g0, . . . , gq−1, q ≥ N, be q meromorphic mappings of Cn into PN (C) with
reduced representations gj = (gj0 : . . . : gjN , 0 6 j 6 q − 1. We say that g0, . . . , gq−1
are located in general position if det(gjkl) 6≡ 0 for any 0 6 j0 < j1 < . . . < jN 6 q−1.
LetMn be the field of all meromorphic functions on Cn. Denote byR
({
gj
}q−1
j=0
)
⊂
⊂Mn the smallest subfield which contains C and all
gjk
gjl
with gjl 6≡ 0.
Let f be a meromorphic mapping of Cn into PN (C) with reduced representation
f = (f0 : . . . : fN ). We say that f is linearly nondegenerate over R
({
gj
}q−1
j=0
)
if
f0, . . . , fN are linearly independent over R
({
gj
}q−1
j=0
)
.
Let ft : Cn → PN (C), 1 6 t 6 λ, be meromorphic mappings with reduced repre-
sentations ft := (ft0 : . . . : ftN ). Let gj : Cn → PN (C), 0 6 j 6 q − 1, be moving
targets located in general position with reduced representations gj := (gj0 : . . . : gjN ).
Assume that (ft, gj) :=
∑N
i=0
ftigji 6= 0 for each 1 6 t 6 λ, 0 6 j 6 q − 1 and
(f1, gj)
−1{0} = . . . = (fλ, gj)
−1{0}. Put Aj = (f1, gj)
−1{0} for each 0 6 j 6 q − 1.
Assume that every analytic set Aj has the irriducible decomposition as follows Aj =
= ∪tji=1Aji, 1 6 tj 6 ∞. Set A = ∪Aji 6≡Akl{Aji ∩ Akl} with 1 6 i 6 tj , 1 6 l 6 tk,
0 6 j, k 6 q − 1.
Denote by T [N+1, q] the set of all injective maps from {1, . . . , N+1} to {0, . . . , q−
− 1}. For each z ∈ Cn \ {∪β∈T [N+1,q]{z|gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) = 0} ∪ A ∪
∪ ∪λi=1I(fi)}, we define ρ(z) = ]{j|z ∈ Aj}. Then ρ(z) 6 N. Indeed, suppose that
z ∈ Aj for each 0 6 j 6 N. Then
∑N
i=0
f1i(z) · gji(z) = 0 for each 0 6 j 6 N. Since
*The research of the authors is supported in part by an NAFOSTED grant of Vietnam.
c© DUC THOAN PHAM, VIET DUC PHAM, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7 923
924 DUC THOAN PHAM, VIET DUC PHAM
gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) 6= 0, it implies that f1i(z) = 0 for each 0 6 i 6 N. This
means that z ∈ I(f1). This is impossible.
For any positive number r > 0, define ρ(r) = sup{ρ(z) | |z| 6 r}, where the
supremum is taken over all z ∈ Cn \
{
∪β∈T [N+1,q] {z|gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) =
= 0} ∪A ∪ ∪λi=1I(fi)
}
. Then ρ(r) is a decreasing function. Let
d := lim
r→+∞
ρ(r).
Then d 6 N. If for each i 6= j, dim{Ai ∩Aj} 6 n− 2, then d = 1.
Theorem A (see [2], Theorem 1). Let f1, . . . , fλ : C → PN (C) be nonconstant
holomorphic curves. Let gi : C → PN (C), 0 6 i 6 q − 1, be moving targets located
in general position and T (r, gi) = o(max16j6λ T (r, fj)), 0 6 i 6 q − 1. Assume
that (fi, gj) 6≡ 0 for 1 6 i 6 λ, 0 6 j 6 q − 1, and Aj := (f1, gj)
−1{0} = . . .
. . . = (fλ, gj)
−1{0} for each 0 6 j 6 q − 1. Denote A = ∪q−1j=0Aj . Let l, 2 6 l 6 λ,
be an integer such that for any increasing sequence 1 6 j1 < . . . < jl 6 λ, fj1(z)∧ . . .
. . . ∧ fjl(z) = 0 for every point z ∈ A. If q >
dN2(2N + 1)λ
λ− l + 1
, then f1, . . . , fλ are
algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C.
Theorem B (see [2], Theorem 2). In addition to the assumption in Theorem A we
assume further that fi, 1 6 i 6 λ, are linearly nondegenerated. Then f1, . . . , fλ are
algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C, if q >
dN(N + 2)λ
λ− l + 1
.
With the same assumption on the nondegeneracy of small moving targets, it is our
main purpose of the present paper to show some algebraic dependence theorems of
meromorphic mappings from Cn into PN (C) for moving targets in more general situ-
ations. Namely, we are going to prove the following.
Theorem 1. Let f1, . . . , fλ : Cn → PN (C) be nonconstant meromorphic map-
pings. Let gi : Cn → PN (C), 0 6 i 6 q − 1, be moving targets located in general
position and T (r, gi) = o(max16j6λ T (r, fj)), 0 6 i 6 q− 1. Assume that (fi, gj) 6≡ 0
for 1 6 i 6 λ, 0 6 j 6 q−1, and Aj := (f1, gj)
−1{0} = . . . = (fλ, gj)
−1{0} for each
0 6 j 6 q − 1. Denote A = ∪q−1j=0Aj . Let l, 2 6 l 6 λ, be an integer such that for any
increasing sequence 1 6 j1 < . . . < jl 6 λ, fj1(z) ∧ . . . ∧ fjl(z) = 0 for every point
z ∈ A. Then f1, . . . , fλ are algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on
C, if q >
dN(2N + 1)λ
λ− l + 1
.
Theorem 2. In addition to the assumption in Theorem 1 we assume further that
fi, 1 6 i 6 λ, are linearly nondegenerate over R
(
{gj}q−1j=0
)
. Then f1, . . . , fλ are
algebraically dependent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C, if q >
dN(N + 2)λ
λ− l + 1
.
Theorem 3. Let f1, . . . , fλ : Cn → PN (C) be nonconstant meromorphic map-
pings. Let gi : Cn → PN (C), 0 6 i 6 q − 1, be moving targets located in general
position such that T (r, gi) = o(max16j6λ T (r, fj)), 0 6 i 6 q−1, and (fi, gj) 6≡ 0 for
1 6 i 6 λ, 0 6 j 6 q − 1. Let κ be a positive integer or κ =∞ and κ = min{κ, N}.
Assume that the following conditions are satisfied:
(i) min{κ, ν(f1,gj)} = . . . = min{κ, ν(fλ,gj)} for each 0 6 j 6 q − 1,
(ii) dim{z|(f1, gi)(z) = (f1, gj)(z) = 0} 6 n− 2 for each 0 6 i < j 6 q − 1,
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ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 925
(iii) there exists an integer number l, 2 6 l 6 λ, such that for any increasing
sequence 1 6 j1 < . . . < jl 6 λ, fj1(z) ∧ . . . ∧ fjl(z) = 0 for every point z ∈
∈ ∪q−1i=0 (f1, gi)
−1{0}.
Then
(i) If q >
N(2N + 1)λ− (κ − 1)(λ− 1)
λ− l + 1
, then f1, . . . , fλ are algebraically depen-
dent over C, i.e., f1 ∧ . . . ∧ fλ ≡ 0 on C;
(ii) if fi, 1 6 i 6 λ, are linearly nondegenerate over R{gj}q−1j=0 and
q >
N(N + 2)λ− (κ − 1)(λ− 1)
λ− l + 1
,
then f1, . . . , fλ are algebraically dependent over C;
(iii) if fi, 1 6 i 6 λ, are linearly nondegenerate over C, gi, 0 6 i 6 q − 1, are
constant mappings and (q − N − 1)((λ − 1)(κ − 1) + q(λ − l + 1)) 6 qNλ, then
f1, . . . , fλ are algebraically dependent over C.
2. Basic notions and auxiliary results from Nevanlinna theory. 2.1. We set
‖z‖ =
(
|z1|2 + . . .+ |zn|2
)1/2
for z = (z1, . . . , zn) ∈ Cn and define
B(r) := {z ∈ Cn : ‖z‖ < r}, S(r) := {z ∈ Cn : ‖z‖ = r}, 0 < r <∞.
Define
vn−1(z) :=
(
ddc‖z‖2
)n−1
and
σn(z) := dclog‖z‖2 ∧
(
ddclog‖z‖2
)n−1
on Cn \ {0}.
2.2. Let F be a nonzero holomorphic function on a domain Ω in Cn. For a set
α = (α1, . . . , αn) of nonnegative integers, we set |α| = α1 + . . . + αn and DαF =
=
∂|α|F
∂α1z1 . . . ∂αnzn
. We define the map νF : Ω→ Z by
νF (z) := max
{
m : DαF (z) = 0 for all α with |α| < m}, z ∈ Ω.
We mean by a divisor on a domain Ω in Cn a map ν : Ω → Z such that, for each
a ∈ Ω, there are nonzero holomorphic functions F and G on a connected neighbourhood
U ⊂ Ω of a such that ν(z) = νF (z)− νG(z) for each z ∈ U outside an analytic set of
dimension 6 n− 2. Two divisors are regarded as the same if they are identical outside
an analytic set of dimension 6 n−2. For a divisor ν on Ω we set |ν| := {z : ν(z) 6= 0},
which is a purely (n− 1)-dimensional analytic subset of Ω or empty.
Take a nonzero meromorphic function ϕ on a domain Ω in Cn. For each a ∈ Ω, we
choose nonzero holomorphic functions F and G on a neighbourhood U ⊂ Ω such that
ϕ =
F
G
on U and dim(F−1(0) ∩G−1(0)) 6 n− 2, and we define the divisors νϕ, ν∞ϕ
by νϕ := νF , ν
∞
ϕ := νG, which are independent of choices of F and G and so globally
well-defined on Ω.
2.3. For a divisor ν on Cn and for a positive integer M or M =∞, we define the
counting function of ν by
ν(M)(z) = min {M,ν(z)},
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n(t) =
∫
|ν| ∩B(t)
ν(z)vn−1 if n ≥ 2,∑
|z|6t
ν(z) if n = 1.
Similarly, we define n(M)(t).
Define
N(r, ν) =
r∫
1
n(t)
t2n−1
dt, 1 < r <∞.
Similarly, we define N(r, ν(M)) and denote them by N (M)(r, ν) respectively.
Let ϕ : Cn → C be a meromorphic function. Define
Nϕ(r) = N(r, νϕ), N (M)
ϕ (r) = N (M)(r, νϕ).
For brevity we will omit the character (M) if M =∞.
2.4. Let f : Cn → PN (C) be a meromorphic mapping. For arbitrarily fixed ho-
mogeneous coordinates (w0 : . . . : wN ) on PN (C), we take a reduced representation
f = (f0 : . . . : fN ), which means that each fi is a holomorphic function on Cn
and f(z) =
(
f0(z) : . . . : fN (z)
)
outside the analytic set {f0 = . . . = fN = 0} of
codimension ≥ 2. Set ‖f‖ =
(
|f0|2 + . . .+ |fN |2
)1/2
.
The characteristic function of f is defined by
T (r, f) =
∫
S(r)
log‖f‖σn −
∫
S(1)
log‖f‖σn.
Let a be a meromorphic mapping of Cn into PN (C) with reduced representation
a = (a0 : . . . : aN ). We define
mf,a(r) =
∫
S(r)
log
‖f‖‖a‖
|(f, a)|
σn −
∫
S(1)
log
‖f‖‖a‖
|(f, a)|
σn,
where ‖a‖ =
(
|a0|2 + . . .+ |aN |2
)1/2
.
If f, a : Cn → PN (C) are meromorphic mappings such that (f, a) 6≡ 0, then the
first main theorem for moving targets in value distribution theory (see [3]) states
T (r, f) + T (r, a) = mf,a(r) +N(f,a)(r).
2.5. Let ϕ be a nonzero meromorphic function on Cn, which are occationally
regarded as a meromorphic map into P1(C). The proximity function of ϕ is defined
by
m(r, ϕ) :=
∫
S(r)
log max (|ϕ|, 1)σn.
2.6. As usual, by the notation ′′‖P ′′ we mean the assertion P holds for all r ∈
∈ [0,∞) excluding a Borel subset E of the interval [0,∞) with
∫
E
dr <∞.
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ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 927
2.7. The First Main Theorem for general position [1, p. 326]. Let fi : Cn →
→ PN (C), 1 6 i 6 λ, be meromorphic mappings located in general position. Assume
that 1 6 λ 6 N + 1. Then
N(r, µf1∧...∧fλ) +m(r, f1 ∧ . . . ∧ fλ) 6
∑
16i6λ
T (r, fi) +O(1).
Let V be a complex vector space of dimension N ≥ 1. The vectors {v1, . . . , vk} are
said to be in general position if for each selection of integers 1 6 i1 < . . . < ip 6 k
with p 6 N, then vi1 ∧ . . . ∧ vip 6= 0. The vectors {v1, . . . , vk} are said to be in special
position if they are not in general position. Take 1 6 p 6 k. Then {v1, . . . , vk} are said
to be in p-special position if for each selection of integers 1 6 i1 < . . . < ip 6 k, the
vectors vi1 , . . . , vip are in special position.
2.8. The Second Main Theorem for general position ([1, p. 320], Theorem 2.1).
Let M be a connected complex manifold of dimension m. Let A be a pure (m − 1)-
dimensional analytic subset of M. Let V be a complex vector space of dimension n+1 >
> 1. Let p and k be integers with 1 6 p 6 k 6 n+ 1. Let fj : M → P (V ), 1 6 j 6 k,
be meromorphic mappings. Assume that f1, . . . , fk are in general position. Also assume
that f1, . . . , fk are in p-special position on A. Then we have
µf1∧...∧fk ≥ (k − p+ 1)νA.
2.9. The Second Main Theorem for moving target. 2.9.1 ([4], Theorem 3.1). Let
f : Cn → PN (C) be a meromorphic mapping. Let {a1, . . . , aq}, q 6= 2, be a set of q
meromorphic mappings of Cn into PN (C) in general position such that f is linearly
nondegenerate over R
({
aj
}q
j=1
)
. Then
q
N + 2
T (r, f) 6
q∑
i=1
N
(N)
(f,ai)
(r) +O
(
max
06i6q−1
T (r, ai)
)
+o
(
T (r, f)
)
.
2.9.2 ([5], Corollary 1) . Let f : Cn → PN (C) be a meromorphic mapping. Let
A = {a1, . . . , aq}, q ≥ 2N+1, be a set of q meromorphic mappings of Cn into PN (C)
located in general position such that (f, ai) 6≡ 0 for each 1 6 i 6 q. Then
q
2N + 1
T (r, f) 6
q∑
i=1
N
(N)
(f,ai)
(r) +O
(
max
16i6q
T (r, ai)
)
+O
(
log+ T (r, f)
)
.
3. Proofs of main theorems. 3.1. Proof of Theorem 1. It suffices to prove Theo-
rem 1 in the case of λ 6 N + 1.
Assume that f1 ∧ . . . ∧ fλ 6≡ 0. We denote by µf1∧...∧fλ the divisor associated with
f1 ∧ . . . ∧ fλ. Denote by N(r, µf1∧...∧fλ) the counting function associated with the
divisor µf1∧...∧fλ . We now prove the following.
Claim 3.1.1. For every 1 6 t 6 λ, we have
q∑
j=1
min{N, ν(ft,gj)(z)} 6
dN
λ− l + 1
µf1∧...∧fλ(z) + qN
∑
β
µgβ(1)∧...∧gβ(N+1)
(z)
for each z 6∈ A∪∪λi=1I(fi), where the sum is over all injective maps β : {1, 2, . . . , N +
+ 1} → {1, 2, . . . , q}.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
928 DUC THOAN PHAM, VIET DUC PHAM
We now prove Claim 3.1.1.
For each regular point z0 ∈ A \ (A ∪ ∪λi=1I(fi) ∪ ∪β∈T [N+1,q]{z|gβ(1)(z) ∧ . . .
. . .∧gβ(N+1)(z) = 0}), let S be an irriducible analytic subset of A containing z0. Since
z0 6∈ A and A = ∪Aij 6≡Akl{Aji ∩ Akl}, where Aji are the irriducible components of
Aj = (f1, gj)
−1{0}, it implies that S is a pure (n− 1)-dimentional analytic subset and
hence, S is only contained in at most d sets of Aj . Thus ν(ft,gj)(z0) 6= 0 at most d
indices. We have
q∑
j=1
min{N, ν(ft,gj)(z0)} 6 dN.
For each increasing sequence 1 6 j1 < . . . < jl 6 λ, we have
fj1(z) ∧ . . . ∧ fjl(z) = 0 ∀z ∈ S.
This implies that the fimily {f1, . . . , fλ} is in l-special position on S. By the Second
Main Theorem for general position [1, p. 320] (Theorem 2.1), we have
µf1∧...∧fλ(z) ≥ (λ− (l − 1))νS .
By the properties of divisor, we have
µf1∧...∧fλ(z0) ≥ λ− l + 1.
Hence
q∑
j=1
min{N, ν(ft,gj)(z0)} 6 dN 6
dN
λ− l + 1
µf1∧...∧fλ(z0).
If z0 ∈ ∪β∈T [N+1,q]{z|gβ(1)(z) ∧ . . . ∧ gβ(N+1)(z) = 0}, then we have
q∑
j=1
min{N, ν(ft,gj)(z0)} 6 qN 6 qN
∑
β∈T [N+1,q]
µgβ(1)∧...∧gβ(N+1)
(z0).
From the above cases and by the properties of divisor, for each z 6∈ A ∪ ∪λi=1I(fi),
we have
q∑
j=1
min{N, ν(ft,gj)(z)} 6
6
dN
λ− l + 1
µf1∧...∧fλ(z) + qN
∑
β∈T [N+1,q]
µgβ(1)∧...∧gβ(N+1)
(z).
Claim 3.1.1 is proved.
The above assertions and The First Main Theorem for general position [1, p. 326],
yield that
q∑
j=1
N
(N)
(ft,gj)
(r) 6
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ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 929
6
dN
λ− l + 1
N(r, µf1∧...∧fλ) + qN
∑
β∈T [N+1,q]
N(r, µgβ(1)∧...∧gβ(N+1)
) 6
6
dN
λ− l + 1
λ∑
i=1
T (r, fi) + qN
∑
β∈T [N+1,q]
N+1∑
i=1
T (r, gβ(i)) =
=
dN
λ− l + 1
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
Thus, by summing them up, we have
λ∑
t=1
q∑
j=1
N
(N)
(ft,gj)
(r) 6
dNλ
λ− l + 1
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
. (1)
By using the Second Main Theorem for moving targets [5] (Corollary 1), it implies
that
λ∑
t=1
q
2N + 1
T (r, ft) 6
dNλ
λ− l + 1
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
Letting r → +∞, we get q 6
dN(2N + 1)λ
λ− l + 1
. This is a contradiction. Thus, the
family {f1, . . . , fλ} is algebraically dependent over Cn, i.e., f1 ∧ . . . ∧ fλ = 0.
Theorem 1 is proved.
3.2. Proof of Theorem 2. From (1), we have
λ∑
t=1
q∑
j=1
N
(N)
(ft,gj)
(r) 6
dNλ
λ− l + 1
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
By using the Second Main Theorem for moving targets [4] (Theorem 3.1), it implies
that
λ∑
t=1
q
N + 2
T (r, ft) 6
dNλ
λ− l + 1
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
Letting r → +∞, we get q 6
dN(N + 2)λ
λ− l + 1
. This is a contradiction. Thus, the
family {f1, . . . , fλ} is algebraically dependent over Cn, i.e. f1 ∧ . . . ∧ fλ = 0.
Theorem 2 is proved.
3.3. Proof of Theorem 3. It suffices to prove Theorem 3 in the case of λ 6 N+1.
Assume that f1 ∧ . . . ∧ fλ 6≡ 0.
We now prove the following.
Claim 3.3.1. For any λ− 1 moving targets gj1 , . . . , gjλ−1
∈ {gj}qj=1, there exists
gj0 6∈ {gj1 , . . . , gjλ−1
} such that
det
(f1, gj1) . . . (fλ, gj1)
...
...
...
(f1, gjλ−1
) . . . (fλ, gjλ−1
)
(f1, gj0) . . . (fλ, gj0)
6≡ 0.
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930 DUC THOAN PHAM, VIET DUC PHAM
We now prove Claim 3.3.1.
Suppose on contrary. Without loss of generality, we assume gj1 = g1, . . . , gjλ−1=gλ−1.
Then
rank
(f1, g1)(z) . . . (fλ, g1)(z)
...
...
...
(f1, gN+1)(z) . . . (fλ, gN+1)(z)
6 λ− 1
for each z ∈ Cn. By f1 ∧ . . . ∧ fλ 6≡ 0, there exists z0 ∈ Cn such that f1(z0) ∧ . . .
. . . ∧ fλ(z0) 6= 0 and z0 6∈ {g1 ∧ . . . ∧ gN+1}−1(0). On the other hand, we have (f1, g1)(z0) . . . (fλ, g1)(z0)
...
...
...
(f1, gN+1)(z0) . . . (fλ, gN+1)(z0)
=
=
g10(z0) . . . g1N (z0)
...
...
...
gN+10(z0) . . . gN+1N (z0)
f10(z0) . . . fλ0(z0)
...
...
...
f1N (z0) . . . fλN (z0)
.
Since the family {gj}qj=1 is located in general position, is implies that the matrix f10(z0) . . . fλ0(z0)
...
...
...
f1N (z0) . . . fλN (z0)
is of rank 6 λ− 1. This is a contradiction.
The Claim 3.3.1 is proved.
We now consider λ − 1 moving targets g1, . . . , gλ−1. Then, by Claim 3.3.1, there
exists gj0 with j0 > λ− 1 such that
det
(f1, g1) . . . (fλ, g1)
...
...
...
(f1, gλ−1) . . . (fλ, gλ−1)
(f1, gj0) . . . (fλ, gj0)
6≡ 0.
Without loss of generality, we may assume that j0 = λ.
Now we putA := ∪λj=1(f1, gj)
−1{0}, A = ∪16i<j6λ
(
(f1, gi)
−1{0}∩(f1, gj)
−1{0}
)
.
We now show the following.
Claim 3.3.2. For each 1 6 t 6 λ, we have
λ∑
i=1
(
min
{
κ, ν(ft,gi)(z)
}
+ (λ− l) min
{
1, ν(ft,gi)(z)
})
+
+
q∑
i=λ+1
(λ− l + 1) min{1, ν(ft,gi)(z)} 6
6 µf̃1∧...∧f̃λ(z) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z) (2)
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ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 931
for every z ∈ Cn \ (A ∪ ∪λi=1I(fi)), where f̃i := ((fi, g1) : . . . : (fi, gλ)) for each
1 6 i 6 λ. Furthermore we have
λ∑
i=1
(
N
(κ)
(ft,gi)
(r) + (λ− l)N (1)
(ft,gi)
(r)
)
+
+
q∑
i=λ+1
(λ− l + 1)N
(1)
(ft,gi)
(r) 6
6
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
. (3)
We now prove Claim 3.3.2.
By the properties of divisor, we only consider three cases for regular points.
Case 1. Let z0 ∈ A\ (A∪∪λi=1I(fi)∪{z|g1∧ . . .∧gλ(z) = 0}) be a regular point
of A. Then z0 is only a zero of one of the meromorphic functions {(ft, gj)}λj=1. Without
loss of generality, we may assume that z0 is a zero of (ft, g1). Let S be an irriducible
analytic subset of A, containing z0. Then the pure dimension of S is n − 1. Suppose
that U is an open neighbourhood of z0 in Cn such that U ∩ {A \ S} = ∅. Choose a
holomorphic function h on Cn such that νh = min{κ, ν(ft,g1)} if z ∈ S and νh = 0
if z 6∈ S. Then (fi, g1) = aih, 1 6 i 6 λ, where ai are holomorphic functions. Since
the matrix
(f1, g2)(z) . . . (fλ, g2)(z)
...
...
...
(f1, gλ)(z) . . . (fλ, gλ)(z)
is of rank 6 λ − 1 for each z ∈ Cn,
it implies that there exist holomorphic functions b1, . . . , bλ such that there is at least
bi 6≡ 0 and
λ∑
i=1
bi(fi, gj) = 0, 2 6 j 6 λ.
Without loss of generality, we may assume that the set of common zeros of {bi}λi=1
is an analytic subset of codimension ≥ 2. Then there exist an index i1, 1 6 i1 6 λ such
that S 6⊂ b−1i1 {0}. We can assume that i1 = λ. Then for each z ∈ (U ∩ S) \ b−1λ {0}, we
have
f̃1(z) ∧ . . . ∧ f̃λ(z) = f̃1(z) ∧ . . . ∧ f̃λ−1(z) ∧
(
f̃λ(z) +
λ−1∑
i=1
bi
bλ
f̃i(z)
)
=
= f̃1(z) ∧ . . . ∧ f̃λ−1(z) ∧ (V (z)h(z)) =
= h(z)
(
f̃1(z) ∧ . . . ∧ f̃λ−1(z) ∧ V (z)
)
,
where V (z) :=
(
aλ +
∑λ−1
i=1
bi
bλ
ai, 0, . . . , 0
)
.
By assumption, for any increasing sequence 1 6 j1 < . . . < jl 6 λ − 1, we have
fj1 ∧ . . . ∧ fjl ≡ 0 on S. Then
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932 DUC THOAN PHAM, VIET DUC PHAM
rank(fj1(z), . . . , fjl(z)) = rank
fj10(z) . . . fjl0(z)
...
...
...
fj1N (z) . . . fjlN (z)
6 l − 1 ∀z ∈ S.
On the other hand (fj1 , g1)(z) . . . (fjl , g1)(z)
...
...
...
(fj1 , gλ)(z) . . . (fjl , gλ)(z)
=
=
g10(z) . . . g1N (z)
...
...
...
gλ0(z) . . . gλN (z)
fj10(z) . . . fjl0(z)
...
...
...
fj1N (z) . . . fjlN (z)
.
Hence
rank
(
f̃j1(z), . . . , f̃jl(z)
)
=
= rank
(fj1 , g1)(z) . . . (fjl , g1)(z)
...
...
...
(fj1 , gλ)(z) . . . (fjl , gλ)(z)
6 l − 1 ∀z ∈ S.
Therefore, f̃j1 ∧ . . . ∧ f̃jl ≡ 0 on S. This implies that the family {f̃1, . . . f̃λ−1} is in
l-special position on S, and {f̃1, . . . f̃λ−1, V } is in (l + 1)-special position on S. By
using The Second Main Theorem for general position [1, p. 320] (Theorem 2.1), we
have
µf̃1∧...∧f̃λ−1∧V (z) ≥ (λ− l)νS ∀z ∈ S.
Hence
µf̃1∧...∧f̃λ(z) ≥ νh(z) + (λ− l)νS =
= min{κ, ν(ft,g1)(z)}+ (λ− l)νS ,∀z ∈ (U ∪ S) \ b−1i1 {0}.
By the properties of divisors, we have
µf̃1∧...∧f̃λ(z0) ≥ min{κ, ν(ft,g1)(z0)}+ λ− l.
This implies that
λ∑
i=1
(
min{κ, ν(ft,gi)(z0)}+ (λ− l) min{1, ν(ft,gi)(z0)}
)
+
+
q∑
i=λ+1
(λ− l + 1) min{1, ν(ft,gi)(z0)} =
= min{κ, ν(ft,g1)(z0)}+ λ− l 6
6 µf̃1∧...∧f̃λ(z0) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z0).
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ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 933
Case 2. Let z0 ∈ A\ (A∪∪λi=1I(fi)∪{z|g1∧ . . .∧gλ(z) = 0}) be a regular point
of A. Then z0 is only a zero of (ft, gi), i > λ. By the assumption, we have the family
{f̃1, . . . , f̃λ} is in l-special position on an irriducible analytic subset of codimension 1
of A which containing z0. By using The Second Main Theorem for general position [1,
p. 320] (Theorem 2.1), we have
µf̃1∧...∧f̃λ(z0) ≥ λ− l + 1.
Hence
λ∑
i=1
(
min{κ, ν(ft,gi)(z0)}+ (λ− l) min{1, ν(ft,gi)(z0)}
)
+
+
q∑
i=λ+1
(λ− l + 1) min{1, ν(ft,gi)(z0)} =
= (λ− l + 1) min{1, ν(ft,gi)(z0)} =
= λ− l + 1 6 µf̃1∧...∧f̃λ(z0) + (λ(κ + λ− l)+
+(q − λ)(λ− l + 1))µg1∧...∧gλ(z0)).
Case 3. Assume that z0 ∈ (g1 ∧ . . . ∧ gλ)−1{0}. Then
λ∑
i=1
(
min{κ, ν(ft,gi)(z0)}+ (λ− l) min{1, ν(ft,gi)(z0)}
)
+
+
q∑
i=λ+1
(λ− l + 1) min{1, ν(ft,gi)(z0)} 6
6 λ(κ + (λ− l)) + (q − λ)(λ− l + 1) 6
6 µf̃1∧...∧f̃λ(z0) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z0)).
From the above cases and by the properties of divisors, for each z 6∈ A ∪λi=1 I(fi),
we have
λ∑
i=1
(
min{κ, ν(ft,gi)(z)}+ (λ− l) min{1, ν(ft,gi)(z)}
)
+
+
q∑
i=λ+1
(λ− l + 1) min{1, ν(ft,gi)(z)} 6
6 µf̃1∧...∧f̃λ(z) + (λ(κ + λ− l) + (q − λ)(λ− l + 1))µg1∧...∧gλ(z).
The first assertion of Claim 3.3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
934 DUC THOAN PHAM, VIET DUC PHAM
By the assumption and definiton of the characteristic function, for each 1 6 j 6 λ,
we have
T (r, f̃j) 6 T (r, fj) + o
(
max
16j6λ
{T (r, fi)}
)
.
By The First Main Theorem for general position [1, p. 326], it implies that
λ∑
i=1
(
N
(κ)
(ft,gi)
(r) + (λ− l)N (1)
(ft,gi)
(r)
)
+
q∑
i=λ+1
(λ− l + 1)N
(1)
(ft,gi)
(r) 6
6 N(r, µf̃1∧...∧f̃λ)(r) +
(
λ(κ + λ− l) + (q − λ)(λ− l + 1)
)
Nµg1∧...∧gλ (r) 6
6
λ∑
i=1
T (r, f̃i) +
(
λ(κ + λ− l) + (q − λ)(λ− l + 1)
) λ∑
i=1
T (r, gi) +O(1) 6
6
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
The second assertion of Claim 3.3.2 is proved.
Thus, for any increasing sequence 1 6 i1 < . . . < iλ−1 6 q, we have
λ−1∑
j=1
(
N
(κ)
(ft,gij )
(r) + (λ− l)N (1)
(ft,gij )
(r)
)
+
∑
i∈I
(λ− l + 1)N
(1)
(ft,gi)
(r) 6
6
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
,
where I = {1, 2, . . . , q} \ {i1, . . . , iλ−1}.
Thus, by summing-up them over all sequences 1 6 i1 < . . . < iλ−1 6 q, we have
q∑
i=1
((λ− 1)N
(κ)
(ft,gi)
(r) + ((λ− 1)(λ− l)+
+(q − λ+ 1)(λ− l + 1))N
(1)
(ft,gi)
(r)) 6
6 q
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
Since κN (N)
f (r) 6 NN
(κ)
f (r) ∀κ, we have
q∑
i=1
(
(λ− 1)κN (N)
(ft,gi)
(r) + ((λ− 1)(λ− l)+
+(q − λ+ 1)(λ− l + 1))N
(N)
(ft,gi)
(r)
)
6
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS . . . 935
6 qN
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
This implies that
q∑
i=1
(
(λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1)
)
N
(N)
(ft,gi)
(r) 6
6 qN
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
Thus, by summing them up over all t (1 6 t 6 λ), we have
q∑
i=1
λ∑
t=1
(
(λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1)
)
N
(N)
(ft,gi)
(r) 6
6 qNλ
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
T (r, fi)
)
. (4)
We now prove the assertions of Theorem 3.
i) By applying the Second Main Theorem for moving targets [5] (Corollary 1) to the
left-hand side of (4), it implies that
q
2N + 1
λ∑
i=1
(
(λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1)
)
T (r, fi) 6
6 qNλ
λ∑
i=1
T (r, fi) + o
(
max
16i6λ
{T (r, fi)}
)
.
Letting r → +∞, we have
q 6 λ− 1 +
(2N + 1)Nλ− (λ− 1)κ − (λ− 1)(λ− l)
λ− l + 1
=
=
(2N + 1)Nλ− (λ− 1)(κ − 1)
λ− l + 1
.
This is a contradiction. Thus, we have f1 ∧ . . . ∧ fλ ≡ 0.
ii) By applying the Second Main Theorem for moving targets [4] (Theorem 3.1) to
the left-hand side of (4), it implies that
q
N + 2
λ∑
i=1
((λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1))T (r, fi) 6
6 qNλ
λ∑
i=1
T (r, fi) + o
(
max
16j6λ
{T (r, fi)}
)
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
936 DUC THOAN PHAM, VIET DUC PHAM
Letting r → +∞, we have
q 6 λ− 1 +
(N + 2)Nλ− (λ− 1)κ − (λ− 1)(λ− l)
λ− l + 1
=
=
(N + 2)Nλ− (λ− 1)(κ − 1)
λ− l + 1
.
This is a contradiction. Thus, we have f1 ∧ . . . ∧ fλ ≡ 0.
iii) By applying the Second Main Theorem for hyperplanes in general position [6,
p. 304] to the left-hand side of (4), it implies that
(q −N − 1)
λ∑
i=1
((λ− 1)κ + (λ− 1)(λ− l) + (q − λ+ 1)(λ− l + 1))T (r, fi) 6
6 qNλ
λ∑
i=1
T (r, fi) + o
(
max
16j6λ
{T (r, fi)}
)
.
Letting r → +∞, we have
(q −N − 1)((λ− 1)(κ − 1) + q(λ− l + 1)) 6 qNλ.
This is a contradiction. Thus, we have f1 ∧ . . . ∧ fλ ≡ 0.
Theorem 3 is proved.
1. Stoll W. On the propagation of dependences // Pacif. J. Math. – 1989. – 139. – P. 311 – 337.
2. Ru M. A uniqueness theorem with moving targets without counting multiplicity // Proc. Amer. Math.
Soc. – 2001. – 129. – P. 2701 – 2707.
3. Ru M., Stoll W. The second main theorem for moving targets // J. Geom. Anal. – 1991. – 1. – P. 99 – 138.
4. Do Duc Thai, Si Duc Quang. Uniqueness problem with truncated multiplicities of meromorphic mappings
in several complex variables for moving targets // Int. J. Math. – 2005. – 16. – P. 903 – 942.
5. Do Duc Thai, Si Duc Quang. Second main theorem with truncated counting function in several complex
variables for moving targets // Forum Math. – 2008. – 20. – P. 145 – 179.
6. Stoll W. Value distribution theory for meromorphic maps // Aspects Math. E. – 1985. – 7.
Received 03.01.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
|
| id | umjimathkievua-article-2925 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:32:55Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/10/a0f6a99af86e229ddc3ae67b8bb37f10.pdf |
| spelling | umjimathkievua-article-29252020-03-18T19:40:27Z Algebraic dependences of meromorphic mappings in several complex variables Алгебраїчна залежність мероморфних відображень для багатьох комплексних змінних Pham, Duc Thoan Pham, Viet Duc Фам, Дік Тон Фам, В'єт Дік We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces. Наведено деякі теореми про алгебраїчну залежність мероморфних відображень для багатьох комплексних змінних на комплексні проективні простори. Institute of Mathematics, NAS of Ukraine 2010-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2925 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 7 (2010); 923–936 Український математичний журнал; Том 62 № 7 (2010); 923–936 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2925/2600 https://umj.imath.kiev.ua/index.php/umj/article/view/2925/2601 Copyright (c) 2010 Pham Duc Thoan; Pham Viet Duc |
| spellingShingle | Pham, Duc Thoan Pham, Viet Duc Фам, Дік Тон Фам, В'єт Дік Algebraic dependences of meromorphic mappings in several complex variables |
| title | Algebraic dependences of meromorphic mappings in several complex variables |
| title_alt | Алгебраїчна залежність мероморфних відображень для багатьох комплексних змінних |
| title_full | Algebraic dependences of meromorphic mappings in several complex variables |
| title_fullStr | Algebraic dependences of meromorphic mappings in several complex variables |
| title_full_unstemmed | Algebraic dependences of meromorphic mappings in several complex variables |
| title_short | Algebraic dependences of meromorphic mappings in several complex variables |
| title_sort | algebraic dependences of meromorphic mappings in several complex variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2925 |
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