NEVANLINNA THEORY FOR MEROMORPHIC MAPS FROM A CLOSED SUBMANIFOLD OF C l TO A COMPACT COMPLEX MANIFOLD

NEVANLINNA THEORY FOR MEROMORPHIC MAPS
FROM A CLOSED SUBMANIFOLD OF Cl TO A
COMPACT COMPLEX MANIFOLD
DO DUC THAI AND VU DUC VIET

Abstract. The purpose of this article is threefold. The first is
to construct a Nevanlina theory for meromorphic mappings from
a polydisc to a compact complex manifold. In particular, we give
a simple proof of Lemma on logarithmic derivative for nonzero
meromorphic functions on Cl . The second is to improve the definition of the non-integrated defect relation of H. Fujimoto [7] and
to show two theorems on the new non-integrated defect relation
of meromorphic maps from a closed submanifold of Cl to a compact complex manifold. The third is to give a unicity theorem
for meromorphic mappings from a Stein manifold to a compact
complex manifold.

Contents
1. Introduction
2. Some facts from pluri-potential theory
2.1. Derivative of a subharmonic function
2.2. Pluri-complex Green function
2.3. Pluri-subharmonic functions on complex manifolds
3. Nevanlinna theory in polydiscs
3.1. First main theorem
3.2. Second main theorem
4. Non-integrated defect relation
4.1. Definitions and basic properties
4.2. Defect relation with a truncation
4.3. Defect relation with no truncation
5. A unicity theorem

2

4
4
6
8
10
11
12
18
18
19
26
31

2000 Mathematics Subject Classification. Primary 32H30; Secondary 32H04,
32H25, 14J70.
Key words and phrases. N -subgeneral position, Non-integrated defect relation,
Second main theorem.
The research of the authors is supported by an NAFOSTED grant of Vietnam
(Grant No. 101.01-2011.29).
1


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DO DUC THAI AND VU DUC VIET

References

37

1. Introduction
To construct a Nevanlinna theory for meromorphic mappings between complex manifolds of arbitrary dimensions is one of the most
important problems of the Value Distribution Theory. Much attention
has been given to this problem over the last few decades and several
important results have been obtained. For instance, W. Stoll [17] introduced to parabolic complex manifolds, i.e manifolds have exhausted
functions on the ones with the same role as the radius function in Cl
and constructed a Nevanlinna theory for meromorphic mappings from
a parabolic complex manifold into a complex projective space. In the
same time, P. Griffiths and J. King [9] constructed a Nevanlinna theory
for holomorphic mappings between algebraic varieties by establishing
special exhausted functions on affine algebraic varieties. There is being
a very interesting problem that is to construct explicitly a Nevanlinna
theory for meromorphic mappings from a Stein complex manifold or a
complete K¨ahler manifold to a compact complex manifold. The first
main aim of this paper is to deal with the above mentioned problem
in a special case when the Stein manifold is a polydisc. In particular,
we give a simple proof of Lemma on logarithmic derivative for nonzero
meromorphic functions on Cl (cf. Proposition 3.7 and Remark 3.8 below).
In 1985, H. Fujimoto [7] introduced the notion of the non-integrated
defect for meromorphic maps of a complete K¨ahler manifold into the
complex projective space intersecting hyperplanes in general position
and obtained some results analogous to the Nevanlinna-Cartan defect
relation. We now recall this definition.
Let M be a complete K¨ahler manifold of m dimension. Let f be a
meromorphic map from M into CP n , µ0 be a positive integer and D
be a hypersurface in CP n of degree d with f (M ) ⊂ D. We denote the
intersection multiplicity of the image of f and D at f (p) by ν(f,D) (p)
and the pull-back of the normalized Fubini-Study metric form on CP n
by Ωf . The non-integrated defect of f with respect to D cut by µ0 is
defined by

[µ ]
δ¯f 0 (D) := 1 − inf{η ≥ 0 : η satisfies condition (∗)}.

Here, the condition (*) means that there exists a bounded nonnegative continuous function h on M with zeros of order not less than
min{ν(f,D) , µ0 } such that dηΩf + ddc log h2 ≥ [min{ν(f,D) , µ0 }], where


NEVANLINNA THEORY
√

3

−1 ¯
dc = 4π
(∂ − ∂) and we mean by [ν] the (1, 1)-current associated with
the divisor ν.
Recently, M. Ru and S. Sogome [16] generalized the above result of
H. Fujimoto for meromorphic maps of a complete K¨ahler manifold into
the complex projective space CP n intersecting hypersurfaces in general
position. After that, T.V. Tan and V.V. Truong [18] generalized successfully the above result of H. Fujimoto for meromorphic maps of a
complete K¨ahler manifold into a complex projective variety V ⊂ CP n
intersecting global hypersurfaces in subgeneral position in V in their
sense. Later, Q. Yan [19] showed the non-integrated defect for meromorphic maps of a complete K¨ahler manifold into CP n intersecting
hypersurfaces in subgeneral position in the original sense in CP n . We
would like to emphasize that, in the results of the above mentioned
authors, there have been two strong restrictions.
• The above mentioned authors always required a strong assumption
(C) that functions h in the notion of the non-integrated defect are
continuous. By this request, their non-integrated defect is still small.
• The above mentioned authors always asked a strong assumption as

follows: (H) The complete K¨ahler manifold M whose universal covering
is biholomorphic to the unit ball of Cl .
Motivated by studying meromorphic mappings into compact complex manifolds in [3] and from the point of view of the Nevanlinna
theory on polydiscs, the second main aim of this paper is to improve
the above-mentioned definition of the non-integrated defect relation of
H. Fujimoto by omiting the assumption (C) (cf. Subsection 4.1 below)
and to study the non-integrated defect for meromorphic mappings from
a Stein manifold without the assumption (H) into a compact complex
manifold sharing divisors in subgeneral position (cf. Theorems 4.3 and
4.7 below). As a direct consequence, we get the following Bloch-Cartan
theorem for meromorphic mappings from Cl to a smooth algebraic variety V in CP m missing hypersurfaces in subgeneral position: a nonconstant meromorphic mapping of Cl into an algebraic variety V of
CP m cannot omit (2N + 1) global hypersurfaces in N -subgeneral position in V . We would like to emphasize that, by using our arguments
and their techniques in [16], [18], [19] we can generalize exactly their
results to meromorphic mappings from a Stein manifold without the
assumption (H) into a smooth complex projective variety V ⊂ CP M
(cf. Remark 4.6 below).
In [8], the author gave a unicity theorem for meromorphic mappings
from a complete K¨ahler manifold satisfying the assumption (H) into
the complex projective space CP n . The last aim of this paper is to give


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DO DUC THAI AND VU DUC VIET

an analogous unicity theorem for meromorphic mappings from a Stein
manifold without the assumption (H) to a compact complex manifold.
2. Some facts from pluri-potential theory
2.1. Derivative of a subharmonic function. In this subsection, we
give an estimation of derivative of a subharmonic function. Firstly, we

recall some definitions.
For R > 0, we consider the ball of radius R as follows:
BR = {x ∈ Rn : |x|< R},
where |x| is the Euclidean norm in Rn .
For R = (R1 , · · · , Rn ) with Rj > 0 for each 1 ≤ j ≤ n, we consider
the polydisc with a radius R as follows:
∆R = {x = (x1 , · · · , xn ) ∈ Rn : |xj |< Rj for each 1 ≤ j ≤ n}.
If R1 = · · · = Rn := R > 0, then the polydisc ∆R is denoted by
∆R := ∆(R).
For x ∈ Rn − {0}, put
E(x) =

(2π)−1 log|x| if n = 2,
−|x|2−n /((n − 2)cn ) if n > 2,

where cn is the area of the unit sphere in Rn . The classical Green
function of BR with pole at x ∈ BR is

2

|x|

E(x − y) − E R ( R|x|2x − y) if x = 0, x = y
GR (x, y) =

1
1
1

− Rn−2

if x = 0.
 (n−2)c
|y|n−2
n
Note that GR (x, y) = GR (y, x). The Poisson kernel of B1 is given by
1
P (x, y) = (1 − |x|2 )|y − x|−n , |y|= 1, |x|< 1.
cn
Theorem 2.1. (Riezs representation formula) Let u be a subharmonic
function ≡ −∞ in the ball BR = {x ∈ Rn : |x|< R}. Take 0 < R < R.
Then
x
u(x) =
GR (x, y)dµ(y) +
u(R y)P
, y dω(y), x ∈ BR ,
R
BR
∂B1
where dµ = ∆u as distributions.
For a proof of this theorem, we refer to [2, Proposition 4.22]. The following has a crucial role in the proof of Proposition 3.7 on Logarithmic
derivative lemma.


NEVANLINNA THEORY

5

Proposition 2.2. Let u be a lower-bounded subharmonic function ≡
−∞ in the ball BR . Assume that u has the derivative a.e in BR . Then

∂u
|
(a)| da ≤ S(n)(R−5 + Rn ) sup |u(x)|
R
|x|≤R
∆( 4n
) ∂xk
for each 1 ≤ k ≤ n, where S(n) is a constant depending only on n.
Proof. By Theorem 2.1 and the hypothesis, we have
∂u
(a) =
∂xk

BR/2

∂GR/2 (a, y)
dµ(y)+
∂xk

u(Ry/2)

a
∂P ( R/2
, y)

∂xk

∂B1

dω(y)

By a direct computation, we get
∂GR/2 (a, y)
ak − yk
cn
=−
+
∂xk
|a − y|n

R
2

n−2

|( R2 )2 a − y|a|2 |2 ak
|( R2 )2 a − y|a|2 |n+2
|a|2 (( R2 )2 ak − yk |a|2 )
−
|( R2 )2 a − y|a|2 |n+2

and

a
∂P ( R/2
, y)

∂xk

=

ak
a
a 2 ak
−2 R/2
| R/2
− y|2 −n(1 − | R/2
| ) R/2
a
cn | R/2
− y|n+2

.

Take a such that |a|≤ R/4. Then, there exists S(n) depending only on
n such that
∂GR/2 (a, y)
1
1
1
|
|≤ S(n)
+ n+3 + 2n+3 ,
n−1
∂xk
|a − y|
R
R
a
∂P ( R/2 , y)

|
|≤ S(n).
∂xk
Therefore, for |a|< R/4,
|

∂u
(a)|≤ S(n)
∂xk

BR/2

1
1
1
+ n+3 + 2n+3 dµ(y)+
n−1
|a − y|
R
R
u(Ry/2)dω(y) .
∂B1

In the Riezs representation formula of u, taking x = 0, we get
u(0) =

G3R/4 (0, y)dµ(y) +
B3R/4

u(3Ry/4)P (0, y)dω(y).

∂B1

Hence
dµ(y) ≤ S(n)Rn−2 sup |u(x)|.
BR/2

|x|≤R


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DO DUC THAI AND VU DUC VIET

For convenience, in this proof, S(n) always stands for a constant depending only n. Therefore,
|

∂u
(a)|≤ S(n)
∂xk

BR/2

1
dµ(y)+(R−5 +R−n−5 +1) sup |u(x)| .
|a − y|n−1
|x|≤R

R
), we obtain
Integrating the above inequality over ∆( 4n

1
S(n)

|
R
)
∆( 4n

∂u
(a)| da ≤
∂xk

dµ(y)
R
)
∆( 4n

BR/2

1
da
|a − y|n−1

(R−5 + R−n−5 + 1) sup |u(x)|da

+
R
∆( 4n
)

≤

|x|≤R

dµ(y)
BR/2

+ (R

−5

B(y,|y|+R/2)

1
da
|a − y|n−1

n

+ R ) sup |u(x)|
|x|≤R

≤ (R

−5

n

+ R ) sup |u(x)|.

|x|≤R

2.2. Pluri-complex Green function. For detailed explosion of this
subsection, one may consult [13, Chapter 6, Section 6.5] and [2, Chapter
1
¯ as
(∂ − ∂)
III, Section 6]. Firstly, we denote d = ∂ + ∂¯ and dc = 4iπ
usual.
Let Ω be a connected open subset of Cn and let a be a point in Ω. If
u is a plurisubharmonic function in a neighborhood of a, we shall say
that u has a logarithmic pole at a if
u(z) − log|z − a|≤ O(1), as z → a,
where |z − a| is the Euclidean norm in Cn . The pluricomplex Green
function of Ω with pole at a is
gΩ,a (z) =
sup{u(z) : u ∈ PSH(Ω, [−∞, 0)) and u has a logarithmic pole at a}.
(It is assumed here that sup ∅ = ∞). We would like to notice that
if V is a plurisubharmonic function, then the ddc V ∧ (ddc gΩ,a )n−1 is
well-defined (see [2, Proposition 4.1]). For r ∈ (−∞, 0], put
−1
gr (z) = max{gΩ,a (z), r}, S(r) = gΩ,a
(r).

Define
µr = (ddc gr (z))n − 1{gΩ,a ≥r} (ddc gΩ,a )n , r ∈ (−∞, 0).


NEVANLINNA THEORY

7

Then the measure µr is supported on S(r) and r → µr is weakly continuous on the left. Denote by µΩ,a the weak-limit of µr as r → 0. We now
consider Ω = ∆R = {(z1 , z2 , · · · , zn ) ∈ Cn : |z1 |< R1 , · · · , |zn |< Rn } is
a polydisc in Cn . For brevity, we will denote the polydisc ∆R by ∆ in
the end of this subsection. Then, we have
g∆,a = max1≤j≤n log|

Rj (zj − aj )
|.
Rj2 − zj a¯j

Moreover, µ∆,a is concentrated on the distinguished boundary ∂ ∆ of
∆ and
n

(1)

dµ∆,a =

Rj2 − |aj |2
dt1 · · · dtn .
it |2
|a
−
R
e
j
j
j=1

Theorem 2.3. Let V be a plurisubharmonic function on an open neighborhood of a polydisc ∆ of Cn . Let g∆,a be a pluricomplex Green function
of ∆ with pole at a = (a1 , · · · , an ) ∈ ∆. Then
n

V
∂∆

Rj2 − |aj |2
dt1 · · · dtn − (2π)n V (a) =
it |2
|a
−
R
e
j
j
j=1
0

ddc V ∧ (ddc g∆,a )n−1 dt
−∞

{g∆,a
Proof. By the Lelong-Jensen formula (see [13, Chapter 6, Section 6.5]
and [2, Chapter III, Section 6]) and the fact that (ddc g∆,a )n = (2π)n δ{a} ,
we obtain for r < 0
r

V (ddc g∆,a )n =

µr (V ) −
{gΩ,a
ddc V ∧ (ddc g∆,a )n−1 dt.
−∞

{g∆,a
It is clear that the right-handed side converges to
0

ddc V ∧ (ddc g∆,a )n−1 dt
−∞

{g∆,a
as r tends to 0. Now suppose that V is continuous. Since the supports
of µr ⊂ ∆ and µr weakly converge to µΩ,a , we get µr (V ) → µΩ,a (V ) as
r tends to 0. In general, by taking a decreasing sequence of continuous plurisubharmonic functions Vn converging to V, we get the desired
equality. Notice that µΩ,a (V ) is finite by (1) and
V (ddc g∆,a )n = (2π)n δ{a} (V ) = (2π)n V (a)
∆

is finite or −∞.


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DO DUC THAI AND VU DUC VIET

2.3. Pluri-subharmonic functions on complex manifolds. This
subsection is devoted to prove a version of [12, Theorem A] in the
case where M is Stein and u is a (not necessary continuous) plurisubharmonic function. Throughout this subsection M will denote an mdimensional closed complex submanifold of Cn and the K¨ahler metric
of M is induced from the canonical one of Cn .
Definition 2.4. Let N be a complex manifold and f be a locally integrable real function in N. We say that f is plurisubharmonic function
(or psh function, for brevity) if ddc f ≥ 0 in the sense of currents.
Lemma 2.5. (see [12, Lemma, p.552]) Let N1 be a K¨ahler manifold
and N2 be a complex manifold. Let g be a holomorphic map of N1 to
N2 . Then for each C 2 -psh function f in N2 , f ◦ g is subharmonic in
N1 .
Lemma 2.6. The volume of M is infinite.
Proof. Take a point a ∈ M. Let BM (a, R) be the ball centered at a
of M and of radius R. Put u = |z − a|. Then u is a psh function on
Cn and hence, it is a subharmonic function on M. Since the K¨ahler
metric on M is induced from the canonical one of Cn , it implies that
BM (a, R) ⊂ B(a, R), where B(a, R) is the usual ball centered at a and
of radius R in Cn . Therefore u ≤ R in BM (a, R). By [12, Theorem A],
we get
1
lim inf 2
u2 d vol = ∞.
r→∞ R
BM (a,R)
From this we deduce that M d vol = ∞.
Proposition 2.7. Let u be a psh function on M and K be a compact
subset of M . For each open subset U of M such that K ⊂ U
M,
∞

there exists a decreasing sequence of C -psh functions uk in U such
that uk converge to u, a.e in U. Moreover, if u is non-negative then uk
is non-negative.
Proof. By [10, Chapter VIII, Theorem 8], there exists a holomorphic
retraction α of an open subset V of Cn containing M to M, i.e α is
holomorphic and α|M = idM . Then u ◦ α is a psh function on V. The
conclusion now is deduced immediately from this fact.
As a direct consequence, we get the following.
Corollary 2.8. Let ξ be an increasing convex function in R. Let u be
a psh function on M. Then ξ ◦ u is a psh function. Specially, if u is
non-negative then up (p ≥ 1) is in the Sobolev space H0 (M ) of degree
0 of M.


NEVANLINNA THEORY

9

Theorem 2.9. Let u be a non-negative psh function on M and p be a
positive number greater than 1. Take a point a ∈ M. Let BM (a, R) (or
B(R) for brevity) be the ball centered at a of M and of radius R. Then
one of the following two statements holds:
(i)
1
lim inf 2
up d vol = ∞.
r→∞ R
BM (a,R)
(ii) u is constant a.e in M.
Proof. Suppose that u is not constant a.e in M and

1
(2)
lim inf 2
up d vol = A < ∞.
r→∞ R
BM (a,R)
Then, there exists a sequence {rj } such that
1
rj2

up d vol = A.
BM (a,rj )

By Proposition 2.7, there is a decreasing sequence uk of C ∞ -nonnegative
functions such that uk is psh in B(rk+2 ) and uk converge to u, a.e in
B(rk+1 ). By the monotone convergence theorem, there exists a subsequence of uk , without loss of generality we may assume that this
subsequence is uk , satisfying
1
1
upj d vol ≤ 2
up d vol + 1.
2
rj B(rj )
rj BM (a,rj )
For each j ≥ 1, let ϕj be a Lipschitz continuous function such that
C
ϕj (x) ≡ 1 on B(a, rj ) and ϕj (x) ≡ 0 in M \B(a, rj+1 ) and gradϕj ≤ r−s
a.e on M, where C is a constant which does not depend on index j (see
[12, Lemma 1]). Put
IjN ( ) =

ϕ2j (u2N + )

p−2
2

||grad uN ||2 d vol, N ≥ j

B(rj+1 )

and IjN = lim →0 IjN ( ). By the proof of [12, Theorem 2.1], IjN < +∞
(by 2) and there exists a constant C = C(p) > 0 such that
(3)

N
N
N
Ij+1
IjN ≤ (Ij+1
)2 ≤ C(Ij+1
− IjN ), 1 ≤ j ≤ N.

The rest of the proof is proceeded as in the one of [12, Theorem 2.4].
For convenience we sketch it here.
It is easy to see that for some j, IjN > 0 for an infinite number of
values of N. Since IjN ≤ IkN for j ≤ k ≤ N, it follows that there exist
an index j0 and a sequence Nk → +∞ such that for each m ≥ j0 , Nk ≥
Nk Nk
Nk
m, Im

> 0. Divide (3) by Ij+1
Ij , m ≤ j ≤ Nk and summing over


10

DO DUC THAI AND VU DUC VIET

Nk
≥ C(Nk − m) for a constant C.
j (from m to Nk ), we obtain 1/Im
Hence,

lim

k→+∞

B(rm )

(u2Nk + )

p−2
2

||grad uNk ||2 d vol = 0.

Now, let ϕ ∈ Co2 (M ) and q be the smallest integer greater than (p −
2)/2. Then,

M

uq+1 ∆ϕd vol = lim uq+1
Nk ∆ϕd vol
k→+∞

= −(q + 1) lim uqNk < grad uNk , grad ϕ >
k→+∞

= 0.
In the other words, ∆u = 0 in the sense of currents. Hence, uq+1 ∈ C ∞
by the regularity theorem (so that ”grad uq+1 ” makes sense). Put X =
grad uq+1 . Then,
||X||2 ϕd vol =
=−

< grad uq+1 , ϕX > d vol
uq+1 div(ϕX)d vol

= − lim

k→+∞

uq+1
Nk div(ϕX)d vol

= (q + 1) lim

k→+∞

uq+1

Nk < grad uNk , ϕX > d vol

= 0.
Therefore, X = 0. That means u is constant, a contradiction.
Corollary 2.10. Let u be a psh function on M. Then
eu d vol = ∞.
M

3. Nevanlinna theory in polydiscs
In Cn , consider a polydisc
∆(a, R) = {(z1 , z2 , · · · , zn ) ∈ Cn : |z1 − a1 |< R1 , · · · , |zn − an |< Rn },
where 0 < R1 , · · · , Rn ≤ ∞. In case of a = 0, we simply denote ∆(a, R)
by ∆R . We now construct definitions in the case where Rj < ∞ for each
j. The construction in the case where Rj = ∞ for some j is similar.


NEVANLINNA THEORY

11

π

3.1. First main theorem. Let L →
− X be a holomorphic line bundle
over a compact complex manifold X and d be a positive integer. Let E
be a C-vector subspace of dimension m + 1 of H 0 (X, Ld ). Take a basis
{ck }m+1
k=1 a basis of E. Put B(E) = ∩σ∈E {σ = 0}. Then
∩1≤i≤m+1 {ci = 0} = B(E)
and

ω = ddc log(|c1 |2 + · · · + |cm+1 |2 )1/d
is well-defined on X \ B(E).
Assume that R = (R1 , · · · , Rn ) and R = (R1 , · · · , Rn ), where Rj >
0 and Rj > 0 for each 1 ≤ j ≤ n. Recall that R < R (R ≤ R resp.) if
0 < Rj < Rj (0 < Rj ≤ Rj resp.) for each 1 ≤ j ≤ n.
As usual, we say that the assertion P holds for a.e r ≤ R if the
assertion P holds for each r ≤ R such that rj is excluded a Borel
subset Ej of the interval [0, Rj ] with Ej ds < ∞ for each 1 ≤ j ≤ n.
Let f be a meromorphic mapping of a polydisc ∆R of radius R =
(R1 , · · · , Rn ) into X such that f (∆R ) ∩ B(E) = ∅. We define the
characteristic function of f with respect to E as follows
Tf (r, E) =
=

1
m(∆r0 )
1
m(∆r0 )

0

f ∗ ω ∧ (ddc g∆r ,a )n−1 ds

dm(a)
−∞

∆r0

g∆r ,a

|g∆r ,a |f ∗ ω ∧ (ddc g∆r ,a )n−1 ,

dm(a)
∆r

∆r0

where r0 < R is fixed, r0 < r ≤ R and dm is the Lebesgue measure in
Cn . Remark that the definition does not depend on choosing basic of
E. Take a section σ of Ls for some s. Let D be its zero divisor. Put
|σ|
||σ||:=
s .
2
(|c1 | + · · · + |cm+1 |2 ) 2d
For 1 ≤ k ≤ ∞, the truncated counting function of f to level k with
respect to D is
[k]

Nf (r, D) =
∆r0

=

0

dm(a)
m(∆r0 )

1

m(∆r0 )

min{[f ∗ D], k} ∧ (ddc g∆r ,a )n−1 ds
−∞

g∆r ,a
|g∆r ,a | min{[f ∗ D], k} ∧ (ddc g∆r ,a )n−1

dm(a)
∆r0

∆r

and the proximity function is
mf (r, D) =

1
m(∆r0 )

dm(a)
∆r0

log
∂∆r

1
dµ∆r ,a .
||σ ◦ f ||2

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DO DUC THAI AND VU DUC VIET

For brevity, we will omit the character [k] if k = ∞. Now, take holomorphic functions f0 , f1 , · · · , fm in ∆R such that (f )0 = f ∗ (c1 )0 , fi+1 =
(f )
fi ci+1
for i ≥ 0. We get a reduced representation (f0 , f1 , · · · , fm ) of
ci (f )
f. By the Lelong-Jensen formula, we get
Theorem 3.1. (First main theorem)
sTf (r, E) = Nf (r, D) + mf (r, D) −

log
∆r0

1
dm(a).
||σ ◦ f ||2

Proposition 3.2.
log(|f0 |2 + · · · + |fm |2 )1/d dt1 dt2 · · · dtn + O(1),

Tf (r, E) =
∂ ∆r

where ∂ ∆r is the distinguished boundary of ∆r . Hence, Tf (r, E) is a
convex increasing function of log ri for each i.
We also have an analogue for Nf (r, D).

Remark 3.3. Put log+ s = max{log s, 0} for all s ≥ 0. Let g be a
meromorphic function on ∆R . Then g is considered as a mapping of
∆R into CP 1 by sending z to [g1 (z), g2 (z)], where g = g1 /g2 . Let H1 be
the hyperplane bundle on CP 1 , D = [0, 1] (the divisor consists of only
one component [0, 1] with coefficient 1). Put
log+ |g(reit )| dt1 · · · dtn .

mg (r) =
∂ ∆r

Then mg (r) = mg (r, [0, 1]) + O(1).
In fact, we have
g1 (reit )
| dt + O(1)
g2 (reit )
∂ ∆r
= mg (r, [0, 1]) + O(1).

log+ |g(reit )| dt =
∂ ∆r

log 1 + |

3.2. Second main theorem. Let g be a meromorphic function on
a polydisc ∆R . For an n-tuple α = (α1 , α2 , · · · , αn ) of non-negative
integers, we put
n

αi , Dα g =

|α|=
i=1

∂ |α| g
.
∂ α1 z1 · · · ∂ αn zn

First of all, we need some auxiliary lemmas.
Lemma 3.4. Let f ∈ L1 (∆R ). Then f ∈ L1 (∂ ∆r ) for e.a r ≤ R. Put
f dt1 · · · dtn (r ≤ R)

f1 (r) =
∂ ∆r


NEVANLINNA THEORY

13

and
f (z) dm(z) (r ≤ R).

f2 (r) =
∆r

Then

∂ n f2
(r) = r1 r2 · · · rn f1 (r) for a.e r ≤ R.
∂r1 ∂r2 · · · ∂rn

Proof. It is known that for a disc ∆r in C and h(z) ∈ L1 (∆r ), we have
d
ds

h(z) dm(z) = s
∆s

h dt
∂∆s

for a.e s ≤ r. The proof now is deduced from the Fubini theorem and
the above formula.
Lemma 3.5. Let φ(r) ≥ 0 be a monotone increasing function for 0 <
r ≤ R. Let δ be a positive real number. Then
1
∂ nφ
(s) ≤ Πnj=1
φ(s)1+δ
∂r1 ∂r2 · · · ∂rn
Rj − sj
for all s ≤ R such that sj does not belong to a set Ej ⊂ [0, Rj ] with
1
dt < ∞ (1 ≤ j ≤ n).
Ej Rj −t
Proof. Put rj = Rj − Rj e−ρj , i.e. ρj = − log(

Rj −rj
)
Rj

≥ 0. Then

∂ nφ
∂ nφ
1
(s) =
(s)Πnj=1
.
∂r1 ∂r2 · · · ∂rn
∂ρ1 ∂ρ2 · · · ∂ρn
Rj − sj
The proof is easily deduced from applying [15, Lemma 1.2.1].
Lemma 3.6. ([11, Lemma 2.4]) Assume that T (r) is a continuous and
increasing function for r0 ≤ r < R0 < ∞ and T (r) ≥ 1. Then we have
T r+
outside a set E0 of r such that

R0 − r
< 2T (r)
eT (r)
1
E0 R0 −s

ds < ∞.

Proposition 3.7. (Lemma on logarithmic derivative) Let g be a meromorphic function in ∆R . Then, there exists a constant C depending only
on n such that
n

m

Dα g
g

(r, [0, 1]) ≤ C log Tg (r, E) +

log
j=1

1
Rj − rj

for all α ∈ Zn+ and for all r ≤ R such that rj does not belong to a set
Ej ⊂ [0, Rj ] with Ej Rj1−s ds < ∞ (1 ≤ j ≤ n).


14

DO DUC THAI AND VU DUC VIET

Proof. We consider the case that |α|= 1. The general case follows easily
from this case. Take r < r . Write g = g1 /g2 . Applying Proposition 2.2
for log|g1 |, log|g2 | we get
−r
∆(a, r 8n
)

|∂ log|g|/∂zk | dm(z) ≤

|log|g1 g2 ||.

sup
|x−a|≤ r

−r
2

Hence, for a ∈ ∆r ,
|

(4)
−r
∆(a, r 8n
)

1
∂g/∂zk
| dm(x) ≤ sup max{log|g1 g2 |, log
}
g
g1 g2
|x−a|≤ r −r
2

n

≤

(5)

2rj
Tf (r , E).
r − rj
j=1 j

By dividing the polydisc ∆r into many polydiscs which have the form
−r
∆(a, r8n
), we obtain
n

rj
∂g/∂zk
|dm(x) ≤ C(n)
|
g
Rj − rj
∆r
j=1

(6)

3

Tf (r , E),

where C(n) is a constant depending only on n. Moreover, by Lemma
3.6 we deduce that
n

∂g/∂zk
rj
|
| dt1 · · · dtn ≤ C(n)
g
Rj − rj
∂ ∆r
j=1

3

Tf3n+1 (r, E)

for all r ≤ R such that rj does not belong to a set Ej ⊂ [0, Rj ] with
1
ds < ∞. Hence,
Ej Rj −s
(7)
∂g/∂zk
log |
| dt1 · · · dtn ≤
g
∂ ∆r

n

+

log

j=1

rj
+ (3n + 1) log Tf (r , E) + O(1)
Rj − rj

for all r ≤ R such that rj does not belong to a set Ej ⊂ [0, Rj ] with
1
ds < ∞. The proof is completed.
Ej Rj −s
Remark 3.8. We can apply the above argument to the characteristic
function in the classical sense of a nonzero meromorphic function g in
Cn to get a simple new proof of Lemma on logarithmic derivative.
i

Now, put ∆(a) = {(z1 , · · · , zn ) : |zk |≤ a for all k} and ∆ (a) =
i
{(z1 , · · · , zn ) : |zk |≤ a for k = i, |zi |= a}. The measure mi on ∆ (a) is
the product of the (n − 1) dimensional Lebesgue measure and the usual
measure on a circle in C.


NEVANLINNA THEORY

15

Proposition 3.9. Let g be a meromorphic function in ∆R . Let p, p
be positive real numbers such that p < p . Assume R = (R0 , · · · , R0 )
and p|α|≤ 1. Then, there exist a constant C depending only on n and
a set E ⊂ [0, R0 ] satisfying E R01−t dt < ∞ such that for all a ∈ [0, R0 ]

outside the set E,
i

α

(i) | Dg g |p is integrable in ∆ (a) with the given measure.
(ii)
n

|

i

i=1

∆ (a)

Dα g
|
g

p

dmi (z) ≤

R03
C
Tf ((a, · · · , a), E)
R0 − a (R0 − a)3

p |α|n

Proof. Let αk , k = 1, 2, · · · , |α| be a sequence of n-tuples satisfying:
α1 = 0, |αk |= |αk−1 |+1, for all k ≥ 2. By the proof of Proposition 3.7
and p < 1,
n

3

Dαk g p|α|
rj
| α
| dm(z) ≤ C(n)
k−1 g
Rj − a
∆(a) D
j=1

(3n+1)p|α|

Tf

1
E0 R0 −s

for all a ≤ R outside a set E ⊂ [0, R0 ] with
|

h(a) =
∆(a)

((a, · · · , a), E)p

ds < ∞. Put

Dα g p
| dm(z) (a ∈ [0, R0 ]).
g

Then
|α|

D αk g p
1
h(a) =
| α
| dm(z) ≤
k−1
g
|α|
∆(a) k=2 D
≤

1
|α|

|α|

n

C(n)
j=1

k=2

rj
Rj − a

|α|

k=2

Dαk g p|α|
| α
| dm(z)
k−1 g
∆(a) D

3
(3n+1)p|α|

Tf

((a, · · · , a), E)p

On the other hand, we have
∂h
=a
∂a

n

i=1

∂h
=a
∂a1

n
i

i=1

∆ (a)

|

Dα g
|
g

p

dmi (z).

From this and Lemma 3.5 we get the desired conclusion.
Now, we need the generalized Wronskian of a meromorphic mapping
which is due to H. Fujimoto [7].
Proposition 3.10. Let F : ∆R → CP m be a linearly non-degenerate
mapping. Assume that F = (F0 , · · · , Fm+1 ) is a reduced representation

of F. Then there exist n-tuples α1 , · · · , αm+1 such that
m(m + 1)
|α1 |+ · · · + |αm+1 |≤
and |αk |≤ m (1 ≤ k ≤ m + 1)
2

.


16

DO DUC THAI AND VU DUC VIET

and the generalized Wronskian of F
Wα1 ,··· ,αm+1 (F ) := det(Dαi Fj : 1 ≤ i, j ≤ m + 1) ≡ 0.
Moreover, for such α1 , · · · , αm+1 , we have
Wα1 ,··· ,αm+1 (F )
F1 · · · Fm+1

m+1

≤
∞

min{(Fi )0 , m}.
i=1

Proof. See [7, Proposition 4.5 and Proposition 4.10].
Let L → X be a holomorphic line bundle over a compact complex
manifold X of dimension n and E be a C−vector subspace of H 0 (X, L)

of dimension m + 1. Let {ck }m+1
k=1 be a basis of E and B(E) be the base
locus of E. Define a mapping Φ : X \ B(E) → CP m by
Φ(x) := [c1 (x) : · · · : cm+1 (x)].
Denote by rankE the maximal rank of Jacobian of Φ on X \ B(E). It
is easy to see that this definition does not depend on choosing a basis
of E. Take σj ∈ H 0 (X, L), Dj = {σj = 0} (1 ≤ j ≤ q). Assume that
N ≥ n and q ≥ N + 1.
Definition 3.11. The hypersurfaces D1 , D2 , · · · , Dq is said to be located in N -subgeneral position with respect to E if for any 1 ≤ i0 <
· · · < iN ≤ q, we have ∩N
j=0 Dij = B(E).
Assume that {Dj } is located in N -subgeneral position with respect
to E. Put u = rankE, b = dimB(E) + 1 if B(E) = ∅ and b = −1
if B(E) = ∅. Assume that u > b. We set σi = 1≤j≤m+1 aij cj , where
aij ∈ C. Define a mapping Φ : X → CP m by
Φ(x) := [c1 (x) : · · · : cm+1 (x)].
It is a meromorphic mapping. Let G(Φ) be the graph of Φ. Define
p1 : G(Φ) → X, p2 : G(Φ) → CP m
by p1 (x, z) = x, p2 (x, z) = z. Since X is compact, p1 and p2 are proper.
m
Hence, Y = Φ(X) = p2 (p−1
1 (X)) is an algebraic variety of CP . Moreover, by definition of rankE, Y is of dimension rankE = u. Denote by
H the hyperplane line bundle of CP m . Put Hi := 1≤j≤m+1 aij zj−1 ,
where [z0 , z1 , · · · , zm ] is the homogeneous coordinate of CP m .
For each K ⊂ Q, put c(K) = rank{Hi }i∈K . We also set
n0 ({Dj }) = max{c(K) : K ⊂ Q with |K|≤ N + 1} − 1,
and
n({Dj }) = max{c(K) : K ⊂ Q} − 1.

NEVANLINNA THEORY

17

Then n({Dj }), n0 ({Dj }) are independent of the choice the C-vector
subspace E of H 0 (X, L) containing σj (1 ≤ j ≤ q). We see that
u ≤ n0 ({Dj }) ≤ n({Dj }) ≤ m.
Proposition 3.12. Let notations be as above. Assume that D1 , · · · , Dq
are in N -subgeneral position with respect to E and q ≥ 2N − u + 2 + b.
u−b
Put kN = 2N −u+2+b, sN = n0 ({Dj }) and tN =
.
n({Dj }) − u + 2 + b
Then, there exist Nochka weights ω(j) for {Dj }, i.e there exist constants ω(j) (j ∈ Q) and Θ satisfying the following conditions:
(i) 0 < ω(j) ≤ Θ ≤ 1 (j ∈ Q) and Θ ≥ tN /kN .
(ii)
j∈Q ω(j) ≥ Θ(q − kN ) + tN .
(iii) Let Ej (j ∈ Q) be arbitrary positive real numbers and R be a
subset of Q with |R|= N + 1. Then, there exist j1 , · · · , jsN +1 in
R such that
∩1≤i≤sN +1 Hji ∩ Y = ∩j∈R Hj ∩ Y
and
ω(j)Ej ≤
j∈R

Eji .
1≤i≤sN +1

Proof. See [3, Section 2].
By repeating the argument in [3], we get an analogous version of the

Second Main Theorem in [3]. We state the following theorem without
its proof.
Theorem 3.13. Let X be a compact complex manifold. Let L → X be
a holomorphic line bundle over X. Fix a positive integer d. Let E be
a C-vector subspace of dimension m + 1 of H 0 (X, Ld ). Put u = rankE
and b = dimB(E) + 1 if B(E) = ∅, otherwise b = −1. Take positive
divisors d1 , d2 , · · · , dq of d. Let σj (1 ≤ j ≤ q) be in H 0 (X, Ldj ) such
d

d
d

that σ1d1 , · · · , σq q ∈ E. Set Dj = (σj )0 (1 ≤ j ≤ q). Assume that
D1 , · · · , Dq are in N -subgeneral position with respect to E and u >
b. Let f : ∆R → X be an analytically non-degenerate meromorphic
mapping with respect to E, i.e f (∆R ) ⊂ supp((σ)) for any σ ∈ E \ {0}
and f (∆R ) ∩ B(E) = ∅. Then, for all r ≤ R such that rj does not
belong to a set Ej ⊂ [0, Rj ] with Ej Rj1−s ds < ∞, we have
q

(q − (m + 1)K(E, N, {Dj }))Tf (r, E) ≤
i=1

1 [mdi /d]
N
(r, Di ) + Sf (r),
di f


18

DO DUC THAI AND VU DUC VIET

where kN , sN , tN are defined as in Proposition 3.12 and
K(E, N, {Dj }) =

kN (sN − u + 2 + b)
.
tN

4. Non-integrated defect relation
4.1. Definitions and basic properties. Let all notations be as in
Section 3. The defect of f with respect to D truncated by k in E is
defined by
[k]
Nf (r, D)
[k]
.
δf,E (D) = lim inf 1 −
r→R
sTf (r, E)
We now assume that f is a meromorphic mapping of a connected
complex manifold M into X such that f (M ) ∩ B(E) = ∅. Let D be
a divisor of H 0 (X, Ls ) for some s > 0. For 0 ≤ k ≤ ∞, denote by
[k]
Df,E the set of real numbers η ≥ 0 such that there exists a bounded
measurable nonnegative function h on M such that
1
ηf ∗ ddc log(|c1 |2 + · · · + |cm+1 |2 )1/d + ddc log h2 ≥ min{k, f ∗ D}
s

in the sense of currents. The non-integrated defect of f with respect to
D in E truncated by k is defined by
[k]
[k]
δ¯f,E (D) := 1 − inf{η : η ∈ Df,E }.

Note that this definition does not depend on choosing a base of E.
Remark 4.1. In the original definition of H. Fujimoto [7], when X =
CP k , L is the hyperplane bundle and s = 1 he required that functions
h
h,
ϕ
are continuous, where ϕ is a holomorphic function in M such that
(ϕ)0 = min{k, f ∗ D}.
By [5, Theorem 1], there exists an open subset U of M such that U
is biholomorphic to a polydisc ∆R and M \ U has a zero measure, i.e
if (V, ϕ) is a local coordinate then ϕ((M \ U ) ∩ V ) is of zero Lebesgue
measure.
Proposition 4.2. We have the following properties of the non-integrated
defect:
[k]
(i) 0 ≤ δ¯ (D) ≤ 1.
f,E

[k]
(ii) δ¯f,E (D) = 1 if f (M ) ∩ D = ∅.
[k]
(iii) δ¯ (D) ≥ 1 − k if f ∗ D ≥ k0 min{f ∗ D, 1}.
f,E

k0


NEVANLINNA THEORY

19

(iv) Denote by fU the restriction of f to U and assume that
lim TfU (r, E) = ∞.

r→R

[k]
[k]
Then 0 ≤ δ¯f,E (D) ≤ δfU ,E (D) ≤ 1.

Proof. The properties (i) and (ii) are evident. To prove (iii), put
h=

k
k0

σ(f )
s
2
(|c1 (f )| + · · · + |cm+1 (f )|2 ) d

and η =

k

.
k0

Since f ∗ D ≥ k0 min{f ∗ D, 1}, we get (iii).
We now prove (iv). Take a holomorphic function ϕ in ∆R such that
[k]
(ϕ) = min{fU∗ D, k}. For η ∈ DfU ,E , put
1

v = η log(|c1 (fU )|2 + · · · + |cm+1 (fU )|2 ) d + log h −

1
log ϕ.
s

Then ddc η ≥ 0 and hence, by Corollary 2.3, we get
0≤
∆r 0

dm(a)
m(∆r0 )

vdµ∆r ,a −
∂ ∆r

∆r0

dm(a)
m(∆r0 )

v(ddc g∆r ,a )n
∆r

s

log(|c1 (fU )|2 + · · · + |cm+1 (fU )|2 ) d dµ∆r ,0 +

=η
−
∂ ∆r

log h dµ∆r ,0
∂ ∆r

∂ ∆r

1
log ϕ dµ∆r ,0 −
s

1 [k]
v dm(a) ≤ ηTfU (r, E) − NfU (r, D) + K,
s
∆r0

where K is a constant, because h is bounded from above. This implies
that
[k]
NfU (r, D)
K

1−
≥1−η+
.
sTfU (r, E)
TfU (r, E)
[k]
[k]
Letting r → R, we obtain δ¯ (D) ≤ δ
(D).
f,E

fU ,E

4.2. Defect relation with a truncation. Now we give the nonintegrated defect with a truncation for meromorphic mappings from
a submanifold of Cl to a compact complex manifold.
Theorem 4.3. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical
K¨ahler form of Cl . Let L → X be a holomorphic line bundle over
a compact manifold X. Fix a positive integer d and let d1 , d2 , · · · , dq
be positive divisors of d. Let E be a C-vector subspace of dimension
m + 1 of H 0 (X, Ld ). Put u = rankE and b = dimB(E) + 1 if B(E) =
∅, otherwise b = −1. Let σj (1 ≤ j ≤ q) be in H 0 (X, Ldj ) such
d

d
d

that σ1d1 , · · · , σq q ∈ E. Set Dj = (σj )0 (1 ≤ j ≤ q). Assume that


20

DO DUC THAI AND VU DUC VIET

D1 , · · · , Dq are in N -subgeneral position with respect to E and u > b.
Let f : M → X be an analytically non-degenerate meromorphic mapping with respect to E, i.e f (M ) ⊂ supp((σ)) for any σ ∈ E \ {0} and
f (M ) ∩ B(E) = ∅. Assume that, for some ρ ≥ 0 and for some basis
{ck }m+1
k=1 of E, there exists a bounded measurable function h ≥ 0 on M
such that
ρf ∗ ddc log(|c1 |2 + · · · + |cm+1 |2 )1/d + ddc log h2 ≥ Ric ω.
Then,
q
[md /d]
δ¯f,E i (Di ) ≤ kN + K (E, N, {Dj }),
i=1

where kN , sN , tN are defined as in Proposition 3.12 and K (E, N, {Dj })
is the constant given in the end of the proof.
d
di

Proof. Put σi = 1≤j≤m+1 aij cj , where aij ∈ C. We define a meromorphic mapping Φ : X → CP m by Φ(x) := [c1 (x) : · · · : cm+1 (x)].
Also since X compact, Y = Φ(X) is an algebraic variety of CP m .
Moreover, by definition of rankE, Y is of dimension rankE = u. Put
F = Φ ◦ f . Since f (M ) ∩ B(E) = ∅ and f is non-degenerate with
respect to E, F is meromorphic and linearly non-degenerate. Denote
by Hm the hyperplane bundle of CP m . Put Hi := 1≤j≤m+1 aij zj−1 ,
where [z0 , z1 , · · · , zm ] is the homogeneous coordinate of CP m . By [5,
Theorem 1], there exists an open subset U of M such that U is biholomorphic to a polydisc ∆R and M \ U has a zero measure. For
convenience, we still denote by f, F their restrictions to U. It is easy

to see that
1
Tf (r, L) = TF (r, Hm ) and Nf (r, Di ) = NF (r, Hi ).
(8)
d
Moreover, we get the following.
[kd /d]
[k]
[kd /d]
• NFk (r, Hi ) = Nf i (r, Di ), δ¯F,Hm (Hi ) = δ¯f,Ei (r, Di ).
• Hj1 ∩ · · · ∩ Hjt ∩ Y ⊂ Φ(B(E)) if Dj1 ∩ · · · ∩ Djt = B(E) .
Put
K1 = {R ⊂ {1, 2, · · · , q + m − u + b + 1} : |R |= rank(R ) = m + 1}.
Without loss of generality we may assume Rj = R∗ (1 ≤ j ≤ n).
From now on, we just consider n-tuples r = (r1 , · · · , rn ) such that
rj = r∗ (1 ≤ j ≤ n). Suppose that
lim sup
r→R

Tf (r, E)
= ∞.
− log(R∗ − r∗ )


NEVANLINNA THEORY

21

Then by Theorem 3.13, we have
q

[md /d]

δf,E i (Dj ) ≤ (m+1)K(E, N, {Dj }) ≤
j=1

kN
(m+1)(sN +m−2(u−b−1)).
tN

By Proposition 4.2, we get
q

kN
[md /d]
(m+1)(sN +m−2(u−b−1)).
δ¯f,E i (Dj ) ≤ (m+1)K(E, N, {Dj }) ≤
t
N
j=1
Hence, we can assume
lim sup
r→R

Tf (r, E)
< ∞.
− log(R∗ − r∗ )

Let α1 , · · · , αm+1 be as in Proposition 3.10. Set l0 = |α1 |+ · · · + |αm+1 |
and take t, p with 0 < l0 t ≤ p < 1. Put
ω(j) − (m + 1)(sN − u + 2 + b) + m − u + 1 + b.

lN =
j∈Q

By Proposition 3.9 and the proof of Theorem A in [3], we get the
following.
Claim 1. Let p be a real positive number such that
m+1

|αi |≤ p

p
i=1

m(m + 1)
< 1.
2

Then there exists a positive constant K such that
n

|W (F )(z)|sN −u+2+b
i

i=1

∆

(r∗ )

q+m−u+b+1
|Hj (F (z))|ω(j) )
j=1

p

||F (z)||plN dmi (z)

(R∗ )3n
≤C
TF (r∗ , L)
(R∗ − r∗ )3n
for each 0 < r∗ < R∗ and r∗ outside a set E satisfying
Claim 2.

p

m(m+1)
2

1
dt
E R∗ −t

< ∞.

[m]

ω(j)(νHj (F ) − νHj (F ) ) ≤ (sN − u + 2 + b)νW (F ) .
1≤j≤q+m−u+b+1

By definition of the non-integrated defect, there exist ηi ≥ 0 (1 ≤
i ≤ q + m − u + b + 1) and a nonnegative functions hi such that
ηi F ∗ ddc log(|z1 |2 + · · · + |zm+1 |2 ) + ddc log h2i ≥ min{m, F ∗ Hi }


22

DO DUC THAI AND VU DUC VIET

[m]
and 1 − ηi ≤ δ¯F,Hm (Hi ) ≤ 1. Take a holomorphic function ϕi in M such
that (ϕi ) = min{m, F ∗ Hi }. Put

ui = Θ log

h2i
+ log(|F1 |2 + · · · |Fm+1 |2 )ηi /2 ,
Ki

where Ki is a constant which is greater than h2i and Θ is the constant
in Proposition 3.12. By the above inequality, we see that ui − Θ log ϕi
is a plurisubharmonic function on M and
eui ≤ ||F ||ηi Θ .
Put
q+m−u+b+1

(1 − ηi )

s=

i=1

and
v := log

|W (F )(z)|sN −u+2+b
q+m−u+b+1
|Hj (F (z))|ω(j)
j=1

q+m−u+b+1

+

ui .
i=1

Since ui − Θ log ϕi (Θ ≥ ω(j)) is a psh function and by virtue of Claim
2, it implies that v is a psh function. On the other hand, by the
hypothesis, there exist ρ > 0 and a nonnegative bounded function h
such that
ρ
ΩF + ddc log h2 ≥ Ric ω,
d
where ω is the K¨ahler
form on M. √
||F ||ρ/d h2
Put w = log K0 det(hij ) , where ω = 2−1 i,j hij dzi ∧ d¯
zj in U , K0 is a
2

constant which is greater than h . Then we have
ew ω n ≤ ||F ||ρ/d dx1 ∧ · · · ∧ dxn
and ω is a psh function. Hence,
ew d vol ≤ ||F ||ρ/d dx1 ∧ · · · ∧ dxn in U,
where d vol stands for the volume form of M with respect to the given
K¨ahler metric. Put
ρ/d
α=
,
lN + Θ(s − q)
χ=

|W (F )(z)|sN −u+2+b
q+m−u+b+1
|Hj (F (z))|ω(j)
j=1


NEVANLINNA THEORY

23

and w1 = w + αv. Then w1 is plurisubharmonic. Hence, ew1 is also
plurisubharmonic. We have
|χ|α ||F ||ρ/d+Θα(q+m−u+b+1−s) dm(z)

ew1 d vol ≤
∆r

∆r

|χ|α ||F ||αlN dm(z)

=
∆r

Suppose α = 3nm(m + 1)α < 1. Then by Claim 1, for each r∗ outside
a set E with E R∗1−s ds < ∞, we have
n
α

αlN

|χ| ||F ||

i

∆ (r∗ )

i=1

(R∗ )3n
dmi (z) ≤ C
TF (r∗ , L)
(R∗ − r∗ )3n

Conbining with the fact that lim supr→R
n
i

|χ|α ||F ||αlN dmi (z) ≤ K1

∆ (r∗ )

i=1

Tf (r,E)
− log(R∗ −r∗ )

1
∗
(R − r∗ )α

α

m(m+1)
2

.

< ∞, we get

log

1
∗
R − r∗

α /4

for some K1 and r∗ ∈ [0, R∗ ) − \E. Varying K1 slightly, we may assume
the above inequality holds for all r∗ ∈ [0, R∗ ) by [6, Proposition 5.5].
From this, we conclude that
R∗
w1

e

d vol ≤ K1

U

0

1
(R∗ − t)α

log

1
R∗ − t

α /4

dt + O(1) < ∞

Combining with the fact that M \ U has zero measure, we get
ew1 d vol < ∞.
M

By Proposition 2.10, we get a contradiction. Hence, 3nm(m + 1)α ≥ 1.
This means
lN
3ρnm(m + 1)
+q+m−u+b+1−
≥ s.
Θd
Θ
Put
tN − (m + 1)(sN − u + 2 + b) + m − u + b + 1
+
Θ
3ρnm(m + 1)
.
Θd
By a direct computation and note that
K (E, N, {Dj }) =

q+m−u+b+1
[md /d]
δ¯f,E i (Di )|< q

|s −
i=1


24

DO DUC THAI AND VU DUC VIET

for > 0 small enough and Θ ≥ tN /kN , and
lN ≥ Θ(q − kN ) + tN − (m + 1)(sN − u + 2 + b) + m − u + b + 1.
we obtain the desired inequality.
In the case where X is the complex projective space, L is the hyperplane bundle of X and Dj are hyperplanes in N -subgeneral position,
we get the following.
Corollary 4.4. Let M be an n-dimensional closed complex submanifold
of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler
form of Cl . Let f : M → CP m be a linear non-degenerate meromorphic
map. Let {Hj } be a family of hyperplanes in N -subgeneral position in
CP m . Denote by Ωf the pull-back of the Fubini-Study form of CP m
by f. Assume that, for some ρ ≥ 0, there exists a bounded measurable
function h ≥ 0 on M such that
ρΩf + ddc log h2 ≥ Ric ω.
Then,
q
[m]
δ¯f,E (Di ) ≤ (2N − m + 1) + 2ρnm(2N − m + 1).
i=1

Corollary 4.5. Let M be an n-dimensional closed complex submanifold
of Cl and ω be its K¨ahler form that is induced from the canonical K¨ahler
form of Cl . Let f : M → CP m be a meromorphic mapping. Denote by
Ωf the pull-back of the Fubini-Study form of CP m by f. Assume that
f satisfies the following two conditions:
(i) Assume that, for some ρ ≥ 0, there exists a bounded measurable
function h ≥ 0 on M such that
ρΩf + ddc log h2 ≥ Ric ω,
(ii) f omits ((2N − m + 2) + 2ρnm(2N − m + 1)) hyperplanes in
N -subgeneral position in CP m .
Then f is linearly degenerate.

Remark 4.6. By using our arguments and their techniques in [16],
[18], [19], we can generalize exactly their results to meromorphic mappings from a Stein manifold without the assumption (H) into a smooth
complex projective variety V ⊂ CP M .
Let D1 , · · · , Dq be hypersurfaces in CP n , where q > n. Also, the
hupersurfaces D1 , · · · , Dq are said to be in general position in CP n if
for every subset {i0 , · · · , in } ⊂ {1, · · · , q},
Di0 ∩ · · · ∩ Din = ∅.


NEVANLINNA THEORY

25

We now can prove the following improvement of [16, Theorem 1.1]).
Theorem 4.3’. Let M be an n-dimensional closed complex submanifold of Cl and ω be its K¨ahler form that is induced from the canonical
K¨ahler form of Cl . Let f : M → CP n be a meromorphic map which
is algebraically nondegenerate (i.e. its image is not contained in any
proper subvariety of CP n ). Denote by Ωf the pull-back of the FubiniStudy form of CP n by f. Let D1 , · · · , Dq be hypersurfaces of degree dj
in CP n , located in general position. Let d = l.c.m.{d1 , · · · , dq } (the
least common multiple of {d1 , · · · , dq }). Assume that, for some ρ ≥ 0,
there exists a bounded continuous function h ≥ 0 on M such that
ρΩf + ddc log h2 ≥ Ric ω.
Then for every

> 0,
q

i=1
2

ρl(l − 1)
[l−1]
δ¯f,Hn (Di ) ≤ (n + 1) + +
,
d

where l ≤ 2n +4n en d2n (nI(
a positive real number x.

−1

))n and I(x) := min{k ∈ N : k > x} for

We now recall the definition of the subgeneral position in the sense
of [18, Theorem 1.2].
Let V ⊂ CP N be a smooth complex projective variety of dimension
n ≥ 1. Let n1 ≥ n and q ≥ 2n1 − n + 1. Hypersurfaces D1 , · · · , Dq in
CP N with V ⊆ Dj for all j = 1, ..., q are said to be in n1 -subgeneral
position in V if the two following conditions are satisfied:
(i) For every 1 ≤ j0 < · · · < jn1 ≤ q, V ∩ Dj0 ∩ · · · ∩ Djn1 = ∅.
(ii) For any subset J ⊂ {1, · · · , q} such that 0 < |J| ≤ n and
{Dj , j ∈ J} are in general position in V and V ∩ (∩j∈J Dj ) = ∅, there
exists an irreducible component σJ of V ∩ (∩j∈J Dj ) with dimσJ =
dim V ∩ (∩j∈J Dj ) such that for any i ∈ {1, · · · , q} \ J, if dim V ∩
(∩j∈J Dj ) = dim V ∩ Di ∩ (∩j∈J Dj ) , then Di contains σJ .
We now can prove the following improvement of [18, Theorem 1.2]).
Theorem 4.3”. Let V ⊂ CP N be a smooth complex projective variety
of dimension n ≥ 1. Let n1 ≥ n and q ≥ 2n1 − n + 1. Let D1 , · · · , Dq
be hypersurfaces in CP N of degree dj , in n1 -subgeneral position in V.
Let d = l.c.m.{d1 , · · · , dq } (the least common multiple of {d1 , · · · , dq }).

Let be an arbitrary constant with 0 < < 1. Set
m := 4dn (2n + 1)(2n1 − n + 1) deg V ·

1

+1,