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

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

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


4

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

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


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

=

|
|≤ 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).


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

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



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
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,
∞

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 ( ) =

> 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

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

|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


12

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).

|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


≥ 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


|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)


∂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

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



((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


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 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
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.



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

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

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 )


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

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


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∗ )



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=

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 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


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


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


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

1

+1,