The second main theorem for meromorphic mappings into a complex projective space

Acta Math Vietnam (2013) 38:187–205DOI 10.1007/s40306-013-0011-6

THE SECOND MAIN THEOREMFOR MEROMORPHIC MAPPINGSINTO A COMPLEX PROJECTIVE SPACE

Do Phuong An · Si Duc Quang · Do Duc Thai

Received: 11 October 2012 / Published online: 8 March 2013© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and SpringerScience+Business Media Singapore 2013

Abstract The main purpose of this article is to show the Second Main Theorem for mero-morphic mappings of Cm into Pn(C) intersecting hypersurfaces in subgeneral position withtruncated counting functions. As an application of the above theorem, we give two unicitytheorems for meromorphic mappings of C

m into Pn(C) sharing few hypersurfaces without

counting multiplicity.

Keywords Holomorphic curves · Algebraic degeneracy · Defect relation · Nochka weight

Mathematics Subject Classification (2000) Primary 32H30 · Secondary 32H04 · 32H25 ·14J70

1 Introduction and main results

Let {Hj }q

j=1 be hyperplanes of CP n. Denote by Q the index set {1,2, . . . , q}. Let N ≥ n

and q ≥ N + 1. We say that the family {Hj }q

j=1 are in N -subgeneral position if for everysubset R ⊂ Q with the cardinality |R| = N + 1

⋂

j∈R

Hj = ∅.

If they are in n-subgeneral position, we simply say that they are in general position.

D.P. An · S.D. Quang · D.D. Thai (�)Department of Mathematics, Hanoi National University of Education, 136 XuanThuy str., Hanoi,Vietname-mail: [email protected]

D.P. Ane-mail: [email protected]

S.D. Quange-mail: [email protected]

mailto:[email protected]



188 D.P. AN ET AL.

Let f : Cm → CP n be a linearly nondegenerate meromorphic mapping and {Hj }q

j=1 behyperplanes in N -subgeneral position in CP n. Then Cartan–Nochka’s second main theorem(see [10, 13]) stated that

‖ (q − 2N + n − 1)T (r, f ) ≤q∑

i=1

N [n](r,div(f,Hi)) + o

(T (r, f )

).

As usual, by notation “‖P ” we mean that the assertion P holds for all r ∈ [0,∞) excludinga Borel subset E of the interval [0,∞) with

∫E

dr < ∞.Cartan–Nochka’s second main theorem plays an extremely important role in Nevanlinna

theory, with many applications to Algebraic or Analytic geometry. Over the last few decades,there have been several results generalizing this theorem to abstract objects. Many con-tributed. We refer readers to the articles [2, 9, 11, 12, 14–19, 21, 22] and the referencestherein for the development of related subjects. We recall some recent results and which arethe best results available at present.

Let f : C → Pn(C) be a holomorphic map. Let f̃ = (f0, . . . , fn) be a reduced represen-

tation of f , where f0, . . . , fn are entire functions on C and have no common zeros. TheNevanlinna–Cartan characteristic function Tf (r) is defined by

Tf (r) = 1

2π

∫ 2π

0log

∥∥f̃(reiθ

)∥∥dθ,

where ∥∥f̃ (z)∥∥ = max

{∣∣f0(z)∣∣, . . . ,

∣∣fn(z)∣∣}.

The above definition is independent, up to an additive constant, of the choice of a re-duced representation of f . Let D be a hypersurface in P

n(C) of degree d . Let Q be thehomogeneous polynomial (form) of degree d defining D. The proximity function mf (r,D)

is defined as

mf (r,D) =∫ 2π

0log

‖f̃ (reiθ )‖d‖Q‖|Q(f̃ )(reiθ )|

dθ

2π,

where ‖Q‖ is the sum of the absolute values of the coefficients of Q. The above definitionis independent, up to an additive constant, of the choice of a reduced representation of f .To define the counting function, let nf (r,D) be the number of zeros of Q(f̃ ) in the disk|z| < r , counting multiplicity. The counting function is then defined by

Nf (r,D) =∫ 2π

0

nf (t,D) − nf (0,D)

tdt + nf (0,D) log r.

Note that

Nf (r,D) =∫ 2π

0log

∣∣Q(f )(reiθ

)∣∣ dθ

2π+ O(1).

The Poisson–Jensen formula implies

First Main Theorem Let f : C → Pn(C) be a holomorphic map, and let D be a hypersur-

face in Pn(C) of degree d . If f (C) ⊂ D, then for every real number r with 0 < r < +∞,

mf (r,D) + Nf (r,D) = dTf (r) + O(1),

where O(1) is a constant independent of r .

THE SECOND MAIN THEOREM 189

In 2004, Min Ru [18] showed a second main theorem for algebraically nondegeneratemeromorphic mappings into P

n(C) sharing hypersurfaces in general position in Pn(C). With

the same assumptions, An–Phuong [1] improved the result of Min Ru by giving an explicittruncation level for counting functions. They proved the following.

Theorem A (An–Phuong [1]) Let f be an algebraically nondegenerate holomorphic mapof C into P

n(C). Let {Qi}q

i=1 be q hypersurfaces in Pn(C) in general position with degQi =

di (1 ≤ i ≤ q). Let d be the least common multiple of the di ’s, d = lcm(d1, . . . , dq). Let0 < ε < 1 and let

L ≥ 2d[2n(n + 1)n(d + 1)ε−1

]n.

Then,

‖ (q − n − 1 − ε)Tf (r) ≤q∑

i=1

1

di

N[L]Qi(f )(r) + o

(Tf (r)

).

Using the above result, M. Dulock and Min Ru [6] gave a uniqueness theorem for mero-morphic mappings sharing a family of hypersurfaces in general position as follows.

Theorem B (Dulock–Ru [6]) Let f and g be algebraically nondegenerate holomorphicmappings of C into P

n(C). Let {Qi}q

i=1 be hypersurfaces in Pn(C) in general position with

degQi = di (1 ≤ i ≤ q). Let d0 = min{d1, . . . , dq}, d = lcm(d1, . . . , dq) and L = 2d[2n(n+1)n(d + 1)]n. Assume that f = g on

⋃q

i=1(ZeroQi(f ) ∪ ZeroQi(g)). Then f = g if q >

n + 1 + 2Lnd0

+ 12 .

However, since the truncation level given in Theorem A is far from the sharp, the numberof hypersurfaces in the uniqueness theorem of Dulock–Min Ru is still big.

As the first steps towards establishing the second main theorems for curvilinear divisorsin subgeneral position in a (nonsingular) complex projective variety, recently, D.T. Do andV.T. Ninh in [5] and G. Dethloff, V.T. Tran and D.T. Do in [4] gave Cartan–Nochka’s secondmain theorem with the truncation for holomorphic curves f : C → V intersecting hypersur-faces located in N -subgeneral position in an arbitrary smooth complex projective variety V .We now state their theorem in [4].

Let N ≥ n and q ≥ N + 1. Let V ⊂ PM(C) be a smooth complex projective variety of

dimension n ≥ 1. Hypersurfaces D1, . . . ,Dq in PM(C) with V ⊆ Dj for all j = 1, . . . , q aresaid to be in N -subgeneral position in V if the two following conditions are satisfied:

(i) For every 1 ≤ j0 < · · · < jN ≤ q , V ∩ Dj0 ∩ · · · ∩ DjN = ∅.(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 ofV ∩ (

⋂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 .

Theorem C (Dethloff–Tran–Do [4]) Let V ⊂ PM(C) be a smooth complex projective vari-ety of dimension n ≥ 1. Let f be an algebraically nondegenerate holomorphic mapping of C

into V . Let D1, . . . ,Dq (V ⊆ Dj ) be hypersurfaces in PM(C) of degree dj , in N -subgeneral

position in V , where N ≥ n and q ≥ 2N − n + 1. Then, for every ε > 0, there exist positiveintegers Lj (j = 1, . . . , q) depending on n,degV,N,dj (j = 1, . . . , q), q, ε in an explicit

190 D.P. AN ET AL.

way such that

∥∥∥∥(q − 2N + n − 1 − ε)Tf (r) ≤q∑

j=1

1

dj

N[Lj ]f (r,Dj ).

We would like to emphasize the following.

(i) The condition (ii) in the above definition on N -subgeneral position of Dethloff–Tan–Thai is hard. Thus, their results may not be very useful and applicable due to this reason.

(ii) In the above-mentioned papers and in other papers (see [3, 7] for instance), either thereis no the truncation levels or the truncation levels obtained depend on the given ε. Whenε goes to zero, the truncation level goes to infinite (so the truncation is totally lost). Themost serious and difficult problem (which is supposed to be extremely hard) is to getthe truncation which is independent of ε.

Motivated by this observation, the main purpose of this paper is to show a second maintheorem for meromorphic mappings of C

m into Pn(C) intersecting hypersurfaces in subgen-

eral position with truncated counting functions. First of all, we give the following.

Definition 1.1 Let N ≥ n and q ≥ N + 1. Let D1, . . . ,Dq be hypersurfaces in Pn(C). The

hypersurfaces D1, . . . ,Dq are said to be in N -subgeneral position in Pn(C) if Dj0 ∩ · · · ∩DjN = ∅ for every 1 ≤ j0 < · · · < jN ≤ q .

If {Di}q

i=1 is in n-subgeneral position then we say that it is in general position.Here is our result.

Theorem 1.2 Let f be an algebraically nondegenerate meromorphic mapping of Cm into

Pn(C). Let {Qi}q

i=1 be hypersurfaces of Pn(C) in N -subgeneral position with degQi =

di (1 ≤ i ≤ q). Let d = lcm(d1, . . . , dq) and M = (n+d

n

)−1. Assume that q > (M+1)(2N−n+1)

n+1 .Then, we have

∥∥∥∥

(q − (M + 1)(2N − n + 1)

n + 1

)Tf (r) ≤

q∑

i=1

1

di

N[M]Qi(f )(r) + o

(Tf (r)

).

In the case of hyperplanes, i.e., di = 1 (1 ≤ i ≤ q), then M = n and Theorem 1.2 givesus the above second main theorem of Cartan–Nochka.

As an application of Theorem 1.2 and by introducing some new techniques, we show thefollowing unicity theorem for meromorphic mappings sharing hypersurfaces in subgeneralposition without counting multiplicity.

Theorem 1.3 Let f and g be algebraically nondegenerate meromorphic mappings ofC

m into Pn(C). Let {Qi}q

i=1 be hypersurfaces in Pn(C) in N -subgeneral position with

degQi = di (1 ≤ i ≤ q). Let d = lcm(d1, . . . , dq) and M = (n+d

n

) − 1. Assume that f = g

on⋃q

i=1(ZeroQi(f ) ∪ ZeroQi(g)). Then f = g if q > (M+1)(2N−n+1)

n+1 .

We would like to emphasize that under the same assumption in the above-mentionedresult of Ru–Dulock, the number of hypersurfaces in Theorem 1.3 is really reduced.

As we know, in Theorem 1.3 and also in all previous unicity theorems, meromorphicmappings always assumed to be agree on inverse images of all q hypersurface targets. In


the last section of this paper, we will show a theorem on algebraic degeneracy for twomeromorphic mappings which are only agree on n + 2 hypersurfaces. Namely, we provethe following.

Theorem 1.4 Let f and g be algebraically nondegenerate meromorphic mappings ofC

m into Pn(C). Let {Qi}q

i=1 be q hypersurfaces in Pn(C) in general position with

degQi = di (1 ≤ i ≤ q). Let d = lcm(d1, . . . , dq), L0 = 2d[2n(n + 1)n(d + 1)]n andd0 = min{d1, . . . , dn+2}. Assume that:

(i) dim(ZeroQi(f ) ∩ ZeroQj(f )) ≤ m − 2 for every 1 ≤ i ≤ n + 2, i < j ≤ q ,(ii) f = g on

⋃n+2i=1 (ZeroQi(f ) ∪ ZeroQi(g)).

(iii) min{νQi(f ),L0} = min{νQi(g),L0} for every n + 3 ≤ i ≤ q .

Assume that q ≥ n+ 2 + 2dL0d0

. Then there are at least [ q−n

2 ] indices n+ 3 ≤ i1 < · · · i[ q−n2 ] ≤

q such that

Qi1(f )

Qi1(g)≡ · · · ≡

Qi[ q−n2 ](f )

Qi[ q−n2 ](g)

. (*)

2 Basic notions and auxiliary results from Nevanlinna theory

2.1 The counting function

We set ‖z‖ = (|z1|2 + · · · + |zm|2)1/2 for z = (z1, . . . , zm) ∈ Cm and define

B(r) := {z ∈ C

m : ‖z‖ < r}, S(r) := {

z ∈ Cm : ‖z‖ = r

}(0 < r < ∞).

Define

vm−1(z) := (ddc‖z‖2

)m−1and

σm(z) := dclog‖z‖2 ∧ (ddclog‖z‖2

)m−1on C

m \ {0}.For a divisor ν on C

m and for a positive integer M or M = ∞, we define the countingfunction of ν by

ν[M](z) = min{M,ν(z)

},

n(t) ={∫

|ν|∩B(t)ν(z)vm−1 if m ≥ 2,

∑|z|≤t ν(z) if m = 1.

Similarly, we define n[M](t).Define

N(r, ν) =∫ r

1

n(t)

t2m−1dt (1 < r < ∞).

Similarly, we define N(r, ν[M]) and denote it by N [M](r, ν).Let ϕ : C

m −→ C be a meromorphic function. Denote by νϕ the zero divisor of ϕ. Define

Nϕ(r) = N(r, νϕ), N [M]ϕ (r) = N [M](r, νϕ).

For brevity we will omit the character [M] if M = ∞.

192 D.P. AN ET AL.

2.2 The characteristic function

Let f : Cm −→ P

n(C) be a meromorphic mapping. For arbitrarily fixed homogeneous coor-dinates (w0 : · · · : wn) on P

n(C), we take a reduced representation f = (f0 : · · · : fn), whichmeans that each fi is a holomorphic function on C

m and f (z) = (f0(z) : · · · : fn(z)) out-side the analytic subset {f0 = · · · = fn = 0} of codimension ≥ 2. Set ‖f ‖ = (|f0|2 + · · · +|fn|2)1/2.

The characteristic function of f is defined by

Tf (r) =∫

S(r)

log‖f ‖σm −∫

S(1)

log‖f ‖σm.

2.3 The proximity function

Let ϕ be a nonzero meromorphic function on Cm, which are occasionally 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

)σm.

The Nevanlinna’s characteristic function of ϕ is defined as follows:

T (r,ϕ) = N 1ϕ(r) + m(r,ϕ).

Then

Tϕ(r) = T (r,ϕ) + O(1).

The function ϕ is said to be small (with respect to f ) if ‖ Tϕ(r) = o(Tf (r)).

2.4 Lemma on logarithmic derivative (see [20, Lemma 3.11])

Let f be a nonzero meromorphic function on Cm. Then

|| m

(r,

Dα(f )

f

)= O

(log+ T (r, f )

) (α ∈ Z

m+).

Repeating the argument as in [8, Proposition 4.5], we have the following:

2.5 Proposition

Let Φ0, . . . ,Φk be meromorphic functions on Cm such that {Φ0, . . . ,Φk} are linearly inde-

pendent over C. Then there exist an admissible set

{αi = (αi1, . . . , αim)

}k

i=0⊂ Z

m+

with |αi | = ∑m

j=1 |αij | ≤ k (0 ≤ i ≤ k) such that the following are satisfied:

(i) {Dαi Φ0, . . . , Dαi Φk}ki=0 is linearly independent over M, i.e., det (Dαi Φj ) ≡ 0.

(ii) det(Dαi (hΦj )) = hk+1 · det(Dαi Φj ) for any nonzero meromorphic function h on Cm.


3 Nochka weights for hypersurfaces in subgeneral position

Let {Qi}q

i=1 be q hypersurfaces in Pn(C) of the common degree d . Assume that each Qi

is defined by a homogeneous polynomial Q∗i ∈ C[x0, . . . , xn]. We regard C[x0, . . . , xn] as a

complex vector space and define

rank{Qi}i∈R = rank{Q∗

i

}i∈R

for every subset R ⊂ {1, . . . , q}. It is easy to see that

rank{Qi}i∈R = rank{Q∗

i

}i∈R

≥ n + 1 − dim

(⋂

i∈R

Qi

).

Hence, if {Qi}q

i=1is in N -subgeneral position, by the above equality, we have

rank{Qi}i∈R ≥ n + 1

for any subset R ⊂ {1, . . . , q} with R = N + 1.Let {Hi}q

i=1 be q hyperplanes in CM passing through the coordinates origin. Assume that

each Hi is defined by the linear equation

aij z1 + · · · + aiMzM = 0,

where aij ∈ C (j = 1, . . . ,M), not all zeros. We define the vector associated with Hi by

vi = (ai1, . . . , aiM) ∈ CM.

For each subset R ⊂ {1, . . . , q}, the rank of {Hi}i∈R is defined by

rank{Hi}i∈R = rank{vi}i∈R.

The family {Hi}q

i=1 is said to be in N -subgeneral position if for any subset R ⊂ {1, . . . , q}with R = N + 1,

⋂i∈R Hi = {0}, i.e., rank{Hi}i∈R = M .

By [13, Lemmas 3.3 and 3.4], we have the following.

Lemma 3.1 Let {Hi}q

i=1 be q hyperplanes in Cn+1 in N -subgeneral position, and assume

that q > 2N − n + 1. Then there are positive rational constants ωi (1 ≤ i ≤ q) satisfyingthe following.

(i) 0 < ωj ≤ 1 ∀i ∈ {1, . . . , q}.(ii) Setting ω̃ = maxj∈Q ωj , one gets

q∑

j=1

ωj = ω̃(q − 2N + n − 1) + n + 1.

(iii) n+12N−n+1 ≤ ω̃ ≤ n

N.

(iv) For R ⊂ Q with 0 < R ≤ N + 1, then∑

i∈R ωi ≤ rank{Hi}i∈R .(v) Let Ei ≥ 1 (1 ≤ i ≤ q) be arbitrarily given numbers. For R ⊂ Q with 0 < R ≤ N + 1,

there is a subset Ro ⊂ R such that Ro = rank{Hi}i∈Ro = rank{Hi}i∈R and∏

i∈R

Eωi

i ≤∏

i∈Ro

Ei.

194 D.P. AN ET AL.

The above ωj are called Nochka weights, and ω̃ the Nochka constant.

Lemma 3.2 Let H1, . . . ,Hq be q hyperplanes in CM (M ≥ 2) passing through the coordi-

nates origin. Let k be a positive integer such that k ≤ M . Then there exists a linear subspaceL ⊂ C

M of dimension k such that L ⊂ Hi (1 ≤ i ≤ q) and

rank{Hi1 ∩ L, . . . ,Hil ∩ L} = rank{Hi1 , . . . ,Hil }for every 1 ≤ l ≤ k and 1 ≤ i1 < · · · < il ≤ q .

Proof We prove the lemma by induction on M (M ≥ k) as follows.• If M = k, by choosing L = CM , we get the desired conclusion of the lemma.• Assume that the lemma holds for every k ≤ M ≤ M0 −1. Now we prove that the lemma

also holds for M = M0.Indeed, we assume that each hyperplane Hi is given by the linear equation

ai1x1 + · · · + aiM0xM0 = 0,

where aij ∈ C, not all zeros and (x1, . . . , xM0) is an affine coordinates system of CM0 . Wedenote the vector associated with Hi by vi = (ai1, . . . , aiM0) ∈ C

M0 \ {0}. For each subsetT of {v1, . . . , vq} satisfying T ≤ k, denote by VT the vector subspace of C

M0 generatedby T . Since dimVT ≤ T ≤ k < M0, VT is a proper vector subspace of CM0 . Then

⋃T VT

is nowhere dense in CM0 . Hence, there exists a nonzero vector v = (a1, . . . , aM0) ∈ C

M0 \⋃T VT . Denote by H the hyperplane of C

M0 defined by

a1x1 + · · · + aM0xM0 = 0.

For each vi ∈ {v1, . . . , vM0}, we have v ∈ V{vi }. Then {v, vi} is linearly independent over C.This follows that Hi ⊂ H . Therefore, H ′

i = Hi ∩ H is a hyperplane of H . Also we see thatdimH = M0 − 1.

By the assumption that the lemma holds for M = M0 − 1, there exists a linear subspaceL ⊂ H of dimension k such that L ⊂ H ′

i (1 ≤ i ≤ q) and

rank{H ′

i1∩ L, . . . ,H ′

il∩ L

} = rank{H ′

i1, . . . ,H ′

il

}

for every 1 ≤ l ≤ k,1 ≤ i1 < · · · < il ≤ q .Since L ⊂ H ′

i , it is easy to see that L ⊂ Hi for each i (1 ≤ i ≤ q). On the otherhand, for every 1 ≤ l ≤ k and 1 ≤ i1 < · · · < il ≤ q , we see that v ∈ V{vi1 ,...,vil

}. Thenrank{vi1 , . . . , vil , v} = rank{vi1 , . . . , vil } + 1. This implies that

rank{H ′

i1, . . . ,H ′

il

} = dimH − dim

(l⋂

j=1

H ′ij

)

= M0 − 1 − dim

(H ∩

l⋂

j=1

Hij

)

= rank{Hi1 , . . . ,Hil ,H } − 1

= rank{vi1 , . . . , vil , v} − 1

= rank{vi1 , . . . , vil }= rank{Hi1 , . . . ,Hil }.


This yields

rank{Hi1 ∩ L, . . . ,Hil ∩ L} = dimL − dim

(L ∩

l⋂

j=1

Hij

)

= dimL − dim

(l⋂

j=1

(H ′

ij∩ L

))

= rank{H ′

i1∩ L, . . . ,H ′

il∩ L

}

= rank{Hi1 , . . . ,Hil }.Then we get the desired linear subspace L in this case.

• By the induction principle, the lemma holds for every M . Hence the lemma is proved. �

Lemma 3.3 Let Q1, . . . ,Qq (q > 2N − n + 1) be hypersurfaces of the common degree d

in Pn(C) located in N -subgeneral position. Then there are positive rational constants ωi

(1 ≤ i ≤ q) satisfying the following:

(i) 0 < ωi ≤ 1, ∀i ∈ {1, . . . , q}.(ii) Setting ω̃ = maxj∈Q ωj , one gets

q∑

j=1

ωj = ω̃(q − 2N + n − 1) + n + 1.

(iii) n+12N−n+1 ≤ ω̃ ≤ n

N.

(iv) For R ⊂ {1, . . . , q} with R = N + 1, then∑

i∈R ωi ≤ n + 1.(v) Let Ei ≥ 1 (1 ≤ i ≤ q) be arbitrarily given numbers. For R ⊂ {1, . . . , q} with R =

N + 1, there is a subset Ro ⊂ R such that Ro = rank{Qi}i∈Ro = n + 1 and

∏

i∈R

Eωi

i ≤∏

i∈Ro

Ei.

Proof We assume that each Qi is given by

∑

I∈Id

aiI xI = 0,

where Id = {(i0, . . . , in) ∈ Nn+10 ; i0 + · · · + in = d}, I = (i0, . . . , in) ∈ Id , xI = x

i00 · · ·xin

n

and aiI ∈ C (1 ≤ i ≤ q, I ∈ Id). Put M = (n+d

n

). Denote by Hi = {(zI1 , . . . , zIM ) ∈

CM ; ∑

Ij ∈IdaiIj zIj = 0} the hyperplane in C

M associated with Qi . Then, for each arbi-trary subset R ⊂ {1, . . . , q} with R = N + 1, we have

dim

(⋂

i∈R

Qi

)≥ n + 1 − rank{Qi}i∈R = n + 1 − rank{Hi}i∈R.

Hence

rank{Hi}i∈R ≥ n + 1 − dim

(⋂

i∈R

Qi

)= n + 1.

196 D.P. AN ET AL.

By Lemma 3.2, there exists a linear subspace L ⊂ CM of dimension n + 1 such that

L ⊂ Hi (1 ≤ i ≤ q) and

rank{Hi1 ∩ L, . . . ,Hil ∩ L} = rank{Hi1 , . . . ,Hil }

for every 1 ≤ l ≤ n + 1 and 1 ≤ i1 < · · · < il ≤ q . Hence, for any subset R ∈ {1, . . . , q} withR = N + 1, since rank{Hi}i∈R ≥ n + 1, there exists a subset R′ ⊂ R with R′ = n + 1 suchthat rank{Hi}i∈R′ = n + 1. This implies that

rank{Hi ∩ L}i∈R ≥ rank{Hi ∩ L}i∈R′ = rank{Hi}i∈R′ = n + 1.

This shows that rank{Hi ∩ L}i∈R = n + 1, since dimL = n + 1. Therefore, {Hi ∩ L}q

i=1 is afamily of hyperplanes in L in N -subgeneral position.

By Lemma 3.1, there exist Nochka weights {ωi}q

i=1 for the family {Hi ∩ L}q

i=1 in L.It is clear that assertions (i)–(iv) are automatically satisfied. Now for R ⊂ {1, . . . , q} withR = N + 1, by Lemma 3.1(v) we have

∑

i∈R

ωi ≤ rank{Hi ∩ L}i∈R = n + 1

and there is a subset Ro ⊂ R such that:

Ro = rank{Hi ∩ L}i∈R0 = rank{Hi ∩ L}i∈R = n + 1,

∏

i∈R

Eωi

i ≤∏

i∈Ro

Ei, ∀Ei ≥ 1 (1 ≤ i ≤ q),

rank{Qi}i∈R0 = rank{Hi ∩ L}i∈R0 = n + 1.

Hence the assertion (v) is also satisfied. The lemma is proved. �

4 Second main theorem with truncated counting functions for degenerate case

Let {Qi}i∈R be a set of hypersurfaces in Pn(C) of the common degree d . Assume that each

Qi is defined by∑

I∈Id

aiI xI = 0,

where Id = {(i0, . . . , in) ∈ Nn+10 ; i0 + · · · + in = d}, I = (i0, . . . , in) ∈ Id , xI = x

i00 · · ·xin

n

and (x0 : · · · : xn) is homogeneous coordinates of Pn(C).

Let f : Cm −→ P

n(C) be an algebraically nondegenerate meromorphic mapping with areduced representation f = (f0 : · · · : fn). We define

Qi(f ) =∑

I∈Id

aiI fI ,

where f I = fi0

0 · · ·f inn for I = (i0, . . . , in). Then f ∗Qi = νQi(f ) as divisors.


Lemma 4.1 Let {Qi}i∈R be a set of hypersurfaces in Pn(C) of the common degree d and let

f be a meromorphic mapping of Cm into P

n(C). Assume that⋂q

i=1 Qi = ∅. Then there existpositive constants α and β such that

α‖f ‖d ≤ maxi∈R

∣∣Qi(f )∣∣ ≤ β‖f ‖d .

Proof Let (x0 : · · · : xn) be homogeneous coordinates of Pn(C). Assume that each Qi is

defined by∑

I∈IdaiI x

I = 0. Set Qi(x) = ∑I∈Id

aiI xI and consider the following function

h = maxi∈R |Qi(x)|‖x‖d

,

where ‖x‖ = (∑n

i=0 |xi |2) 12 .

We see that the function h does not depend on the choice of the reduced representationof f and it is a positive continuous function on P

n(C). By the compactness of Pn(C), there

exist positive constants α and β such that α = minx∈Pn(C) h and β = maxx∈Pn(C) h. Therefore,we have

α‖f ‖d ≤ maxi∈R

∣∣Qi(f )∣∣ ≤ β‖f ‖d .

The lemma is proved. �

Lemma 4.2 Let {Qi}q

i=1 be a set of q hypersurfaces in Pn(C) of the common degree d . Put

M = (n+d

n

) − 1. Then, there exist (N − n) hypersurfaces {Ti}M−ni=1 in P

n(C) such that forany subset R ∈ {1, . . . , q} with R = rank{Qi}i∈R = n + 1, then rank{{Qi}i∈R ∪ {Ti}M−n

i=1 } =M + 1.

Proof For each i (1 ≤ i ≤ q), take a homogeneous polynomial Q∗i ∈ C[x0, . . . , xn] of de-

gree d defining Qi . Denote by Hd the complex vector subspace of the complex vectorspace C[x0, . . . , xn] generated by all homogeneous polynomials of degree d . Then dimHd =M + 1.

For each subset R ∈ {1, . . . , q} with R = rank{Q∗i }i∈R = n + 1, denote by VR the

set of all vectors v = (v1, . . . , vM−n) ∈ (Hd)M−n such that {{Q∗

i }i∈R, v1, . . . , vM−n} is lin-early dependent over C. It is clear that VR is an algebraic subset of (Hd)

M−n. SincedimHd = M + 1 and rank{Q∗

i }i∈R = n + 1, there exists v = (v1, . . . , vN−n) ∈ (Hd)M−n such

that {{Q∗i }i∈R, v1, . . . , vM−n} is linearly independent over C, i.e., v ∈ VR . Therefore VR is a

proper algebraic subset of (Hd)M−n for each R. This implies that

(Hd)M−n

∖⋃

R

VR = ∅.

Hence, there is (T ∗1 , . . . , T ∗

M−n) ∈ (Hd)M−n \ ⋃

R VR .Take the hypersurface Ti in P

n(C), which is defined by the homogeneous polynomial T ∗i

(i = 1, . . . , q). We have

rank{{Qi}i∈R ∪ {Ti}N−n

i=1

} = rank{{

Q∗i

}i∈R

∪ {T ∗

i

}M−n

i=1

} = M + 1

for every subset R ∈ {1, . . . , q} with R = rank{Qi}i∈R = n + 1.The lemma is proved. �

198 D.P. AN ET AL.

Proof of Theorem 1.2 We first prove the theorem for the case where all Qi (i = 1, . . . , q)

have the same degree d .It is easy to see that there is a positive constant β such that β‖f ‖d ≥ |Qi(f )| for every

1 ≤ i ≤ q . Set Q := {1, . . . , q}. Let {ωi}q

i=1 be as in Lemma 3.3 for the family {Qi}q

i=1. Let{Ti}M−n

i=1 be (M − n) hypersurfaces in Pn(C) which satisfy Lemma 4.2. Let R ⊂ Q be such

that R = M + 1. Choose Ro ⊂ R such that Ro = rank{Qi}i∈Ro = n + 1 and Ro satisfiesLemma 3.3(v) with respect to numbers { β‖f ‖d

|Qi(f )| }q

i=1. Assume that R := {r1, . . . , rM+1},Ro :={ro

1 , . . . , ron+1}.

We write Id = {I0, . . . , IM}, where Ii = (ti0, . . . , tin) ∈ Id , and set f Ii = fti00 · · ·f tin

n .Since f is algebraically nondegenerate over C, {f Ii }M

i=0 is linearly independent over C.Then there is an admissible set {α0, . . . , αM} ⊂ Z

m+ such that

W ≡ det(

Dαj f Ii (0 ≤ i ≤ M))

0≤j≤M≡ 0

and |αj | ≤ M,∀0 ≤ j ≤ M .Set

WRo ≡ det(

Dαj Qr0k(f )(1 ≤ k ≤ n + 1), Dαj Tl(f )(1 ≤ l ≤ M − n)

)0≤j≤M

.

Since rank{Qr0k(1 ≤ k ≤ n+1), Tl(1 ≤ l ≤ M −n)} = M +1, there exist a nonzero constant

CRo such that WRo = CRo · W .Let z be a fixed point. Then there exists R ⊂ Q with R = N + 1 such that |Qi(f )(z)| ≤

|Qj(f )(z)|,∀i ∈ R,j ∈ R. Since⋂

i∈R Qi = ∅, by Lemma 4.1 there exists a positive con-stant αR such that

αR‖f ‖d(z) ≤ maxi∈R

∣∣Qi(f )(z)∣∣.

Then we see that

‖f (z)‖d(∑q

i=1 ωi )|W(z)||Qω1

1 (f )(z) · · ·Qωqq (f )(z)| ≤ |W(z)|

αq−N−1R βN+1

∏

i∈R

(β‖f (z)‖d

|Qi(f )(z)|)ωi

≤ AR

|W(z)| · ‖f ‖d(n+1)(z)∏i∈Ro |Qi(f )|(z)

≤ BR

|WRo(z)| · ‖f ‖d(M+1)(z)∏

i∈Ro |Qi(f )|(z)∏M−n

i=1 |Ti(f )|(z) ,

where AR,BR are positive constants.Put SR = BR

|WRo |∏

i∈Ro |Qi(f )|∏M−ni=1 |Ti (f )| . By the lemma on logarithmic derivative, it is easy to

see that∥∥∥∥

∫

S(r)

log+ SR(z)σm = o(Tf (r)

).

Therefore, for each z ∈ Cm, we have

log

( ‖f (z)‖d(∑q

i=1 ωi )|W(z)||Qω1

1 (f )(z) · · ·Qωqq (f )(z)|

)≤ log

(‖f ‖d(M+1)(z)) +

∑

R⊂Q,R=N+1

log+ SR.


Integrating both sides of the above inequality over S(r) with the note that∑q

i=1 ωi = ω̃i(q −2N + n − 1), we have

∥∥∥∥ d

(q − 2N + n − 1 − M − n

ω̃

)Tf (r) ≤

q∑

i=1

ωi

ω̃NQi(f )(r)

− 1

ω̃NW(r) + o

(Tf (r)

). (1)

Claim∑q

i=1 ωiNQi(f )(r) − NW(r) ≤ ∑q

i=1 ωiN[M]Qi(f )(r).

Indeed, let z be a zero of some Qi(f )(z) and z ∈ I (f ) = {f0 = · · · = fn = 0}. Since{Qi}q

i=1 is in N -subgeneral position, z is not a zero of more than N functions Qi(f ).Without loss of generality, we may assume that z is a zero of Qi(f ) (1 ≤ i ≤ k ≤ N)

and z is not a zero of Qi(f ) with i > N . Put R = {1, . . . ,N + 1}. Choose R1 ⊂ R withR1 = rank{Qi}i∈R1 = n + 1 such that R1 satisfies Lemma 3.3(v) with respect to numbers{emax{νQi (f )(z)−M,0}}q

i=1. Then, we have∑

i∈R

ωi max{νQi(f )(z) − M,0

} ≤∑

i∈R1

max{νQi(f )(z) − M,0

}.

This yields

νW (z) = νWR1 (z)

≥∑

i∈R1

max{νQi(f )(z) − M,0

}

≥∑

i∈R


}.

Thusq∑

i=1

ωiνQi(f )(z) − νW (z) =∑

i∈R

ωiνQi(f )(z) − νW (z)

=∑

i∈R

ωi min{νQi(f )(z),M

}

+∑

i∈R


} − νW (z)

≤∑

i∈R


}

=q∑

i=1


}.

Integrating both sides of this inequality, we getq∑

i=1

ωiNQi(f )(r) − NW(r) ≤q∑

i=1

ωiN[M]Qi(f )(r).

This proves the claim.

200 D.P. AN ET AL.

Combining the claim and (1), we obtain∥∥∥∥ d

(q − 2N + n − 1 − M − n

ω̃

)Tf (r) ≤

q∑

i=1

ωi

ω̃N

[M]Qi(f )(r) + o

(Tf (r)

)

≤q∑

i=1

N[M]Qi(f )(r) + o

(Tf (r)

).

Since ω̃ ≥ n+12N−n+1 , the above inequality implies that∥∥∥∥ d

(q − (M + 1)(2N − n + 1)

n + 1

)Tf (r) ≤

q∑

i=1

N[M]Qi(f )(r) + o

(Tf (r)

).

Hence, the theorem is proved in the case where all Qi have the same degree.We now prove the theorem for the general case where degQi = di . Applying the above

case for f and the hypersurfaces Qddi

i (i = 1, . . . , q) of the common degree d , we have∥∥∥∥

(q − (M + 1)(2N − n + 1)

n + 1

)Tf (r) ≤ 1

d

q∑

i=1

N[M]Q

d/dii

(f )(r) + o

(Tf (r)

)

≤q∑

i=1

1

d

d

di

N[M]Qi(f )(r) + o

(Tf (r)

)

=q∑

i=1

1

di

N[M]Qi(f )(r) + o

(Tf (r)

).

The theorem is proved. �

5 Unicity of meromorphic mappings sharing hypersurfaces in subgeneral position

Lemma 5.1 Let f and g be nonconstant meromorphic mappings of Cm into Pn(C). LetQi (i = 1, . . . , q) be hypersurfaces in P

n(C) in N -subgeneral position with degQi =di , N ≥ n. Put d = lcm(d1, . . . , dq) and M = (

n+d

n

) − 1. Assume that ZeroQi(f ) =ZeroQi(g) (1 ≤ i ≤ q) and q > (M+1)(2N−n+1)

n+1 . Then ‖ Tf (r) = O(Tg(r)) and ‖ Tg(r) =O(Tf (r)).

Proof Using Theorem 1.2 for f , we have∥∥∥∥

(q − (M + 1)(2N − n + 1)

n + 1

)Tf (r) ≤

q∑

i=1

1

di

N[M]Qi(f )(r) + o

(Tf (r)

)

≤q∑

i=1

M

di

N[1]Qi(f )(r) + o

(Tf (r)

)

=q∑

i=1

M

di

N[1]Qi(g)(r) + o

(Tf (r)

)

≤ qM Tg(r) + o(Tf (r)

).

Hence ‖ Tf (r) = O(Tg(r)). Similarly, we get ‖ Tg(r) = O(Tf (r)). �


Proof of Theorem 1.3 We assume that f and g have reduced representations f = (f0 : · · · :fn) and g = (g0 : · · · : gn), respectively. Replacing Qi by Q

ddi

i if necessary, without loss ofgenerality, we may assume that di = d for all i = 1, . . . , q .

By Lemma 5.1, we have ‖ Tf (r) = O(Tg(r)) and ‖ Tg(r) = O(Tf (r)). Suppose thatf = g. Then there exist two indices s, t (0 ≤ s < t ≤ n) satisfying

H := fsgt − ftgs ≡ 0.

By the assumption of the theorem, we have H = 0 on

q⋃

i=1

(ZeroQi(f ) ∪ ZeroQi(g)

).

Also since {Qi}q

i=1 is in N -subgeneral position, there are at most N functions Qi(f ) van-ishing at each point of

⋃q

i=1(ZeroQi(f ) ∪ ZeroQi(g)). Therefore, we have

ν0H ≥ 1

N

q∑

i=1

min{1, ν0

Qi(f )

}

outside an analytic subset of codimension at least two. Then, it follows that

NH (r) ≥ 1

N

q∑

i=1

N[1]Qi(f )(r). (2)

On the other hand, by the definition of the characteristic function and by the Jensenformula, we have

NH (r) =∫

S(r)

log |fsgt − ftgs |σm

≤∫

S(r)

log‖f ‖σm +∫

S(r)

log‖g‖σm

= Tf (r) + Tg(r).

Combining this and (2), we obtain

Tf (r) + Tg(r) ≥ 1

N

q∑

i=1

N[1]Qi(f )(r).

Similarly, we have

Tf (r) + Tg(r) ≥ 1

N

q∑

i=1

N[1]Qi(g)(r).

Summing up both sides of the above two inequalities, we have

2(Tf (r) + Tg(r)

) ≥ 1

N

q∑

i=1

N[1]Qi(f )(r) + 1

N

q∑

i=1

N[1]Qi(g)(r). (3)

202 D.P. AN ET AL.

From (3) and applying Theorem 1.2 for f and g, we have

2(Tf (r) + Tg(r)

) ≥q∑

i=1

1

NMN

[M]Qi(f )(r) +

q∑

i=1

1

NMN

[M]Qi(g)(r)

≥ d

NM

(q − (M + 1)(2N − n + 1)

n + 1

)

× (Tf (r) + Tg(r)

) + o(Tf (r) + Tg(r)

).

Letting r −→ +∞, we get

2 ≥ d

NM

(q − (M + 1)(2N − n + 1)

n + 1

).

This implies that q ≤ 2NMd

+ (M+1)(2N−n+1)

n+1 . This is a contradiction. Hence f = g. Thetheorem is proved. �

Proof of Theorem 1.4 Suppose that the assertion (*) does not hold. By Lemma 5.1, we have

‖ Tf (r) = O(Tg(r)

)and ‖ Tg(r) = O

(Tf (r)

).

By changing indices if necessary, we may assume that

Q

ddn+3n+3 (f )

Q

ddn+3n+3 (g)

≡ · · · ≡ Q

ddk1k1

(f )

Q

ddk1k1

(g)︸︷︷︸

group 1

≡ Q

ddk1+1

k1+1 (f )

Q

ddk1+1

k1+1 (g)

≡ · · · ≡ Q

ddk2k2

(f )

Q

ddk2k2

(g)︸︷︷︸

group 2

≡ Q

ddk2+1

k2+1 (f )

Q

ddk2+1

k2+1 (g)

≡ · · · ≡ Q

ddk3k3

(f )

Q

ddk3k3

(g)︸︷︷︸

group 3

≡ · · · ≡ Q

ddks−1+1

ks−1+1 (f )

Q

ddks−1+1

ks−1+1 (g)

≡ · · · ≡ Q

ddks

ks(f )

Q

ddks

ks(g)

︸︷︷︸group s

,

where ks = q .Since the assertion (*) does not hold, the number of elements of each group is at most

[ q−n−22 ]. For each n + 3 ≤ i ≤ q , we set

σ(i) ={

i + [ q−n−22 ] if i + [ q−n−2

2 ] ≤ q,

i + [ q−n−22 ] − q + n + 2 if i + [ q−n−2

2 ] > q

and

Pi = Qddi

i (f )Q

ddσ(i)

σ (i) (g) − Qddi

i (g)Q

ddσ(i)

σ (i) (f ).


ThenQ

ddii

(f )

Q

ddii

(g)

andQ

ddσ(i)σ (i)

(f )

Q

ddσ(i)σ (i)

(g)

belong to two distinct groups and hence, Pi ≡ 0 for every n+3 ≤

i ≤ q . Then Pi ≡ 0 (n + 3 ≤ i ≤ q). It is easy to see that

νPi(z) ≥ min

{ν

Q

ddii

(f )

(z), νQ

ddii

(g)

(z)} + min

{ν

Q

ddσ(i)σ (i)

(f )

(z), νQ

ddσ(i)σ (i)

(g)

(z)}

+n+2∑

j=1

min{νQj (f )(z),1

}

= d

di

min{νQi(f )(z), νQi (g)(z)

} + d

dσ(i)

min{νQσ(i)(f )(z), νQσ(i)(g)(z)

}

+n+2∑

j=1

min{νQj (f )(z),1

}

≥ d

di

min{νQi(f )(z), l0

} + d

dσ(i)

min{νQσ(i)(f )(z), l0

} +n+2∑

j=1

min{νQj (f )(z),1

}

for all z outside an analytic subset

S =⋃

1≤i1<···<ik+1≤n+1

(k+1⋂

j=1

ZeroQij (f )

)

of codimension at least 2 of Cm.

Integrating both sides of this inequality, we get

‖ NPi(r) ≥

n+2∑

j=1

N[1]Qj (f ) +

q∑

j=i,σ (i)

d

dj

N[L0]Qj (f ). (4)

Repeating the same argument as in the proof of Theorem 1.3, by Jensen’s formula and bythe definition of the characteristic function, we have

‖ NPi(r) ≤ d

(Tf (r) + Tg(r)

). (5)

From (4) and (5), we get

‖ d(Tf (r) + Tg(r)

) ≥n+2∑

j=1

N[1]Qj (f ) +

q∑

j=i,σ (i)

d

dj

N[L0]Qj (f ).

Summing up both sides of this inequality over all n + 3 ≤ i ≤ q , we obtain

d(q − n − 2)(Tf (r) + Tg(r)

) ≥ (q − n − 2)

n+2∑

j=1

N[1]Qj (f ) + 2

q∑

i=n+3

d

di

N[L0]Qi(f )

≥ 2d

d0

n+2∑

j=1

N[L0]Qj (f ) + 2

q∑

i=n+3

d

di

N[L0]Qi(f )

≥ 2q∑

j=1

d

dj

N[L0]Qj (f ).

204 D.P. AN ET AL.

Choose ε ∈ (0,1) close enough to 1 such that

L0 = 2d[2n(n + 1)n(d + 1)ε−1

]n.

By Theorem A, the above inequality implies that

‖ d(q − n − 2)(Tf (r) + Tg(r)

) ≥ 2d(q − n − 1 − ε)Tf (r) + o(Tf (r)

).

Similarly, we have

‖ d(q − n − 2)(Tf (r) + Tg(r)

) ≥ 2d(q − n − 1 − ε)Tg(r) + o(Tf (r)

).

Therefore,

‖ d(q − n − 2)(Tf (r) + Tg(r)

) ≥ d(q − n − 1 − ε)(Tf (r) + Tg(r)

) + o(Tf (r)

).

Letting r −→ +∞, we get q − n − 2 ≥ q − n − 1 − ε. This means that 1 ≤ ε. This is acontradiction. Hence the assertion (*) holds. The theorem is proved. �

Acknowledgements The research of the authors is supported by an NAFOSTED grant of Vietnam (GrantNo. 101.01-2011.29).

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The second main theorem for meromorphic mappings into a complex projective space