Multiple values and uniqueness problem of meromorphic mappings ...

arXiv:1306.3568v1 [math.CV] 15 Jun 2013

MULTIPLE VALUES AND UNIQUENESS PROBLEM OF MEROMORPHIC MAPPINGS SHARING HYPERSURFACES TING-BIN CAO AND HONG-ZHE CAO Abstract. The purpose of this article is to deal with the multiple values and uniqueness problem of meromorphic mappings from Cm into the complex projective space Pn (C) sharing fixed and moving hypersurfaces. We obtain several uniqueness theorems which improve and extend some known results.

1. Introduction and main results In 1926, R. Nevanlinna [10] proved the well-known five-value theorem that for two nonconstant meromorphic functions f and g on the complex plane C, if they have the same inverse images (ignoring multiplicities) for five distinct values in P1 (C), then f = g. We know that the number five of distinct values in Nevanlinna’s fivevalue theorem cannot be reduced to four. For example, f (z) = ez and g(z) = e−z share four values 0, 1, −1, ∞ (ignoring multiplicities), but f (z) 6≡ g(z). Fujimoto [7] generalized the Nevanlinna’s well-known five-value theorem to the case of meromorphic mappings from Cm into Pn (C), and obtained that for two linearly nondegenerate meromorphic mappings f, g of Cm into Pn (C), if they have the same inverse images of 3n + 2 hyperplanes counted with multiplicities in Pn (C) in general position, then f = g. In 1983, Smiley [16] considered meromorphic mappings which share 3n + 2 hyperplanes of Pn (C) without counting multiplicity and proved the following result. Theorem 1.1. [16] Let f, g be linearly nondegenerate meromorphic mappings of Cm into Pn (C). Let {Hj }qj=1 (q ≥ 3n + 2) be hyperplanes in Pn (C) in generate position. Assume that (a) dim(f −1 (Hi ) ∩ f −1 (Hj )) ≤ m − 2 for all 1 ≤ i < j ≤ q, (b)f (z) = g(z) on ∪qj=1 f −1 (Hj ). (c)f −1 (Hj ) = g −1 (Hj ) for all 1 ≤ j ≤ q, Then f = g. Dulock and Ru [6] gave a slight improvement result as follows: Theorem 1.2. [6] Let f, g be linearly nondegenerate meromorphic mappings of Cm into Pn (C). Let {Hj }qj=1 (q ≥ 3n + 2) be hyperplanes in Pn (C) in generate position. Assume that (a) dim(f −1 (Hi ) ∩ f −1 (Hj )) ≤ m − 2 for all 1 ≤ i < j ≤ q, 2010 Mathematics Subject Classification. Primary 32H30, Secondary 32A22, 30D35. Key words and phrases. Meromorphic mapping, uniqueness theorem, Nevanlinna theory, hypersurfaces. This research was partly supported by the NSFC (no.11101201), the NSF of Jiangxi (no.20122BAB211001) and NSF of education department of Jiangxi(no.GJJ13077) of China. 1

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(b)f (z) = g(z) on ∪qj=1 ((f −1 (Hj ) ∪ g −1 (Hj )). Then f = g. It is natural to extend this results to meromorphic mappings of Cm into Pn (C) sharing hypersurfaces. It seems that Dulock and Ru firstly studied this topic. They proposed that Theorem 1.3. [6] Let {Qj }qj=1 be hypersurfaces of degree deg Qj = dj in Pn (C) in general position. Let d0 = min{d1 , . . . , dq }, d be the least common multiple of dj ’s, i. e., d = lcm(d1 , . . . , dq ), and let M = 2d[2n−1 (n + 1)nd(d + 1)]n . Suppose that f, g are algebraically nondegenerate meromorphic mappings of Cm into Pn (C) such that (a) dim(f −1 (Qi ) ∩ f −1 (Qj )) ≤ m − 2 for all 1 ≤ i < j ≤ q, (b)f (z) = g(z) on ∪qj=1 (f −1 (Qj ) ∪ g −1 (Qj )). If 2M 1 q > (n + 1) + + , d0 2 then f = g. Let V be a complex projective subvariety of Pn (C) of dimension k (k ≤ n). Let d be a positive integer. We denote by I(V ) the ideal of homogeneous polynomials in C[x0 , . . . , xn ] defining V, Hd the ring of all homogeneous polynomials in C[x0 , . . . , xn ] of degree d (which is also a vector space). Define Id (V ) :=

Hd I(V ) ∩ Hd

and HV (d) := dim Id (V ).

Then HV (d) is called Hilbert function of V. Each element of IV (d) which is an equivalent class of an element Q ∈ Hd , will denoted by [Q]. Let f be a meromorphic mapping of Cm into V. We say that f is degenerate over Id (V ) if there is [Q] ∈ Id (V ) \ {0} so that Q(f ) ≡ 0, otherwise we say that f is nondegenerate over Id (V ). This implies that if f is algebraically nondegenerate then f is nondegenerate over Id (V ) for every d ≥ 1. The family of hypersurfaces {Qj }qj=1 is said to be in N −subgeneral position with respect to V if for any 1 ≤ i1 < · · · < iN +1 , +1 V ∩ (∩N j=1 Qij ) = ∅.

If {Qj }qj=1 is said to be in n−subgeneral position with respect to V, then we say that it is in general position with respect to V. Recently, Quang [14] improved Theorem 1.3, the uniqueness theorems in [12], and obtained the following uniqueness theorem for meromorphic mappings which share 3n+2 hypersurfaces of Pn (C) without counting multiplicity, which generalizes Theorem 1.2. Theorem 1.4. [14] Let V be a complex projective subvariety of Pn (C) of dimension k (k ≤ n). Let {Qj }qj=1 be hypersurfaces in Pn (C) in N −subgeneral position with respective to V, deg Qj = dj (1 ≤ j ≤ q). Let d be the least common multiple of dj ’s, i. e., d = lcm(d1 , . . . , dq ). Let f, g be meromorphic mappings of Cm into V which are nondegenerate over Id (V ). Assume that (a) dim(f −1 (Qi ) ∩ f −1 (Qj )) ≤ m − 2 for all 1 ≤ i < j ≤ q,

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(b)f (z) = g(z) on ∪qj=1 (f −1 (Qj ) ∪ g −1 (Qj )). If 2(HV (d) − 1) (2N − k + 1)HV (d) + , q> d k+1 then f = g. Quang [15] also proposed a uniqueness theorem for meromorphic mappings sharing slow moving hypersurfaces without counting multiplicity, which improved the uniqueness result due to Dethloff and Tan [5] and [11]. Theorem 1.5. [15] Let f and g be nonconstant meromorphic mappings of Cm into Pn (C). Let {Qj }qj=1 be set of slow (with respect to f and g) moving hypersurfaces in Pn (C) in weakly general position with degree deg Qj = dj . Put d be  the leastcommon n+d multiple of d1 , . . . , dn+2 , i. e., d = lcm(d1 , . . . , dn+2 ), and N = − 1. let n k (1 ≤ k ≤ n) be an integer. Assume that (a) dim(∩kj=0 f −1 (Qij )) ≤ m − 2 for every 1 ≤ i0 < · · · < ik ≤ q, (b)f (z) = g(z) on ∪qj=1 (f −1 (Qj ) ∪ g −1 (Qj )). Then the following assertions hold: (i) If q > 2kN (nNd +n+1) then f = g. (ii)In addition to the assumptions (a) and (b), we assume further that both f and ˜ {Q }q . If q > 2kN (N +2) then f = g. g are algebraically nondegenerate over K j j=1 d It is interesting to considering multiple values and uniqueness problem for meromorphic mappings. For example, H. X. Yi ([19], Theorem 3.15) adopted the method of dealing with multiple values due to L. Yang [18] and obtained a uniqueness theorem of meromorphic functions of one variable, which generalized the famous Nevanlinna’s five-value theorem: Theorem 1.6. [19, Theorem 3.15] Let f and g be two nonconstant meromorphic functions on C, let aj (j = 1, 2, . . . , q) be q distinct complex elements in P1 (C) and take mj ∈ Z+ ∪ {∞} (j = 1, 2, . . . , q)Psatisfying m1 ≥ m2 ≥ · · · ≥ mq and q mj 1 νf1−aj ,≤mj = νg−a (j = 1, 2, . . . , q). If j=3 mj +1 > 2, then f (z) ≡ g(z). j ,≤mj

Later, Hu, Li and Yang extended this result to meromorphic functions in several variables (see [8, Theorem 3.9]). In 2000, Aihara[1] generalized Theorem 1.6 to the case of meromorphic mappings sharing hyperplanes from Cm into Pn (C). Recently, Cao-Yi[3], L¨ u, Tu-Wang[17], Quang[13], Cao-Liu-Cao[2] continued to investigate this topic. The main purpose of this paper is to consider multiple values and uniqueness problem for meromorphic mappings sharing fixed or moving hypersurfaces from Cm into Pn (C), and generalize Theorem 1.4 and Theorem 1.5 as follows. Theorem 1.7. Let V be a complex projective subvariety of Pn (C) of dimension k (k ≤ n). Let {Qj }qj=1 be hypersurfaces in Pn (C) in N −subgeneral position with respective to V, deg Qj = dj (1 ≤ j ≤ q). Let d be the least common multiple of dj ’s, i. e., d = lcm(d1 , . . . , dq ). Let f, g be meromorphic mappings of Cm into V which are nondegenerate over Id (V ). Let mj (j = 1, · · · , q) be positive integers or ∞ with m1 ≥ · · · ≥ mq ≥ n. Assume that (a) dim(f −1 (Qi ) ∩ f −1 (Qj )) ≤ m − 2 for all 1 ≤ i < j ≤ q,

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(b)f (z) = g(z) on If

∪qj=1 ({z ∈ Cm : 0 < νQj (f ) (z) ≤ mj } ∪ {z ∈ Cm : 0 < νQj (g) (z) ≤ mj }). q X j=3

mj d(2N − k + 1)HV (d) qd 4 − 2HV (d) >q+ − − , mj + 1 (k + 1)(HV (d) − 1) HV (d) − 1 m2 + 1

then f = g. Remark 1.1. (a). Obviously, Theorem 1.4 is just the special case when m1 = m2 = · · · = mq = ∞ in Theorem 1.7. (b). In the case of mapping into Pn (C) sharing hyperplanes in general position, that means V = Pn (C), di = 1(1 ≤ i ≤ q), HV (d) = n + 1, N = n = k, then the condition q X d(2N − k + 1)HV (d) qd 4 − 2HV (d) mj >q+ − − mj + 1 (k + 1)(HV (d) − 1) HV (d) − 1 m2 + 1 j=3 becomes

q X j=3

mj (n − 1)q n + 1 2(n − 1) > + + mj + 1 n n m2 + 1

which implies that this is a slight improvement of Theorem Furthermore, for Pq 1.4m[3]. j the special case when n = 1, the condition reduces to j=3 mj +1 > 2 which is just the condition of Theorem 1.6. Hence, the above theorem generalizes Theorem1.6 and Theorem 3.9 in [8]. (c). Both Theorem 1.4 and Theorem 1.7 imply that q ≥ 3n + 2 in the case of mapping into Pn (C) sharing hyperplanes in general position. However, Chen and Yan [4] obtained that q ≥ 2n + 3 in this case. Thus, there maybe exist a better condition than the conditions on the number q in Theorem 1.4 and Theorem 1.7. Theorem 1.8. Let f and g be nonconstant meromorphic mappings of Cm into Pn (C). Let {Qj }qj=1 be set of slow (with respect to f and g) moving hypersurfaces in Pn (C) in weakly general position with degree deg Qj = dj . Put d be  the leastcommon n+d multiple of d1 , . . . , dn+2 , i. e., d = lcm(d1 , . . . , dn+2 ), and N = − 1. let n k (1 ≤ k ≤ n) be an integer. Let mj (j = 1, · · · , q) be positive integers or ∞ with m1 ≥ · · · ≥ mq ≥ n. Assume that (a) dim(∩kj=0 f −1 (Qij )) ≤ m − 2 for every 1 ≤ i0 < · · · < ik ≤ q, (b)f (z) = g(z) on ∪qj=1 ({z ∈ Cm : 0 < νQj (f ) (z) ≤ mj } ∪ {z ∈ Cm : 0 < νQj (g) (z) ≤ mj }). Then P the following assertions hold: q mj qd −2 > q + 2k − 2 − (nN +n+1)N − 2kN (i) If j=3 mj +1 m2 +1 then f = g. (ii)In addition to the assumptions (a) and (b), we assume further that both f and Pq mj ˜ {Q }q . If g are algebraically nondegenerate over K j=3 mj +1 > q + 2k − 2 − j j=1 qd (N +2)N

−

2kN −2 m2 +1 )

then f = g.

It is easy to see that Theorem 1.5 is just the special case when m1 = m2 = · · · = mq = ∞ in Theorem 1.8.

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2. Preliminaries 2.1. Set kzk = (|z1 |2 + · · · + |zm |2 ) for z = (z1 , · · · , zm ), for r > 0, define Bm (r) := {z ∈ Cm : kzk < r}, Sm (r) := {z ∈ Cm : kzk = r}. √ Let d = ∂ + ∂, dc = (4π −1)−1 (∂ + ∂). Write σm (z) := (ddc kzk2)m−1 ,

for z ∈ Cm \ {0}.

ηm (z) := dc log kzk2 ∧ (ddc kzk2)m−1

m m 2.2. P∞ Let ϕ(6≡ 0) be an entire function on C . For a ∈ C , we write ϕ(z) = P (z − a), where the term P (z) is a homogeneous polynomial of degree i. i i=0 i We denote the zero-multiplicity of ϕ at a by νϕ (a) = min {i : Pi 6= 0}. Set |νϕ | := {z ∈ Cm : νϕ (z) 6= 0}, which is a purely (m − 1)-dimensional analytic subset or empty set. Let h be a nonzero meromorphic function on Cm . For each a ∈ Cm , we choose nonzero holomorphic functions h0 , h1 on a neighborhood U of a such that h = h0 : −1 ∞ h1 on U and dim(h−1 0 (0) ∩ h1 (0)) ≤ m − 2, we define νh := νh0 , νh := νh1 , which are independent of the choice of h0 , h1 .

2.3. Let f : Cm → Pn (C) be a nonconstant meromorphic mapping, (ω0 : · · · : ωn ) be an arbitrarily fixed homogeneous coordinates on Pn (C). We choose holomorphic functions f0 , · · · , fn on Cm such that If := {z ∈ Cm : f0 (z) = · · · = fn (z) = 0}

is of dimension ≤ m − 2, and f = (f0 : · · · : fn ) is called a reduced representation P 1 of f. Set kf k = ( nj=0 |fj |2 ) 2 . The characteristic function of f is defined by Z Z Tf (r) = log kf kηm − log kf kηm (r > 1). Sm (r)

Sm (1)

Note that Tf (r) is independent of the choice of the reduced representation of f. 2.4. For a divisor ν on Cm and let k, M be positive integers or ∞, we define the following counting functions of ν by:  0, if ν(z) > k; M M ν (z) = min{ν(z), M }, ν≤k (z) = ν M (z), if ν(z) ≤ k, ( R ν(z)σm , if m ≥ 2; |ν|∩B(t) P n(t) = if m = 1. |z|≤t ν(z),

M Similarly, we define nM (t), nM >k (t) and n≤k (t). Define Z r n(t) N (r, ν) = dt (1 < r < ∞). 2m−1 1 t M M Similarly, we define N (r, ν M ), N (r, ν>k ) and N (r, ν≤k ) and denote them by N M (r, ν), M M N>k (r, ν) and N≤k (r, ν) respectively. For a meromorphic function f on Cm , we denote by

Nf (r) = N (r, νf ), M M Nf,≤k (r) = N≤k (r, νf ),

NfM (r) = N M (r, νf ), M M Nf,>k (r) = N>k (r, νf ).

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In addition, if M = ∞, we will omit the superscript M for brevity. On the other hand we have the following Jensen’s formula: Z Z Nf (r) − N f1 (r) = log |f |ηm − log |f |ηm . Sm (r)

Sm (1)

n

2.5. Let Q be a hypersurfaces in P (C) of the degree d defined by X aI xI = 0, I∈Id

where Id = {(i0 , . . . , in ) ∈ Nn+1 ; i0 + · · · + in = d}, I = (i0 , . . . , in ) ∈ Id , xI = 0 i0 in x0 · · · xn and (c0 : · · · : cn ) is homogeneous coordinates of Pn (C). let f : Cm → V ⊂ Pn (C) be an algebraically nondegenerate meromorphic mapping into V with a reduced representation f = (f0 : · · · : fn ). We define X Q(f ) = aI f I , I∈Id

f0i0

where f I = · · · fnin for I = (i0 , . . . , in ). We denote by f ∗ Q = νQ(f ) as divisors. We define Nevanlinna’s deficiency δf (Q) by δf (Q) = 1 − lim sup r→∞

NQ(f ) (r) . Tf (r)

If δf (Q) > 0, then Q is called a deficient hypersurface in the sense of Nevanlinna. As usual, ”kP ” means the assertion R P holds for all r ∈ [0, ∞) excluding a Borel subset E of the interval [0, ∞) with E dr < ∞. Theorem 2.1. [14](The Second Main Theorem for fixed hypersurfaces in subgeneral position) Let V be a complex projective subvariety of Pn (C) of dimension k(k ≤ n). Let {Qi }qi=1 be hypersurfaces of Pn (C) in N −subgeneral position with respect to V, with deg Qi = di (1 ≤ i ≤ q). Let d be the least common multiple of di ’s, i.e., d = lcm(d1 , . . . , dq ). Let f be a meromorphic mapping of Cm into V which is V (d) nondegenerate over Id (V ). If q > (2N −k+1)H , then we have k+1   q X 1 [HV (d)−1] (2N − k + 1)HV (d) Tf (r) ≤ (r) + o(Tf (r)). NQi (f ) k q− k+1 d i=1 i

2.6. We denote by M (resp. Kf ) the field of all meromorphic functions (resp. small meromorphic functions) on Cm . Denote by HCm the ring of all holomorphic functions on Cm . Let Q be a homogeneous polynomial in HCm [x0 , . . . , xn ] of degree d ≥ 1. Denote by Q(z) be the homogeneous polynomial over C obtained by substituting a specific point z ∈ Cm into the coefficients of Q. We also call a moving hyper surface in Pn (C) each homogeneous polynomial Q ∈ HCm [x0 , . . . , xn ] such that the common zero set of all coefficients of Q has codimension at least two. Let Q be a moving hypersurface in Pn (C) of degree d given by X aI w I , Q(z) = I∈Id

Nn+1 ; i0 + · · · + in 0

in i0 = d}, aI ∈ HCm and wI =  w · · · w . ′ n+d We consider the meromorphic mapping Q : Cm → PN (C), where N = − n

where Id = {(i0 , . . . , in ) ∈

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1, given by ′

(Id = {I0,...,IN }). Q (z) = (aI0 (z) : · · · : aIN (z)) The moving hypersurface Q is said to be ”slow” (with respect to f ) if kTQ′ (r) = o(Tf (r)). This is equivalent to kT aIi (r) = o(Tf (r)) for every aIj 6≡ 0. aI j

Let {Qi }qi=1 be a family of moving hypersurfaces in Pn (C), deg Qi = di . Assume that X Qi = aiI wI . I∈Idi

˜ {Q }q the smallest subfield of M which contains C and all aiI We denote by K j j=1 aiJ with aiJ 6≡ 0. We say that {Qi }qi=1 are in weakly general position if there exits z ∈ Cm such that all aiI (1 ≤ i ≤ q, I ∈ I) are holomorphic at z and for any 1 ≤ i0 < · · · < in ≤ q the system of equations  Qij (z)(w0 , . . . , wn ) = 0 1≤j≤n has only the trivial solution w = (0, . . . , 0) in Cn+1 .

Theorem 2.2. [15] (The Second Main Theorem for moving hypersurfaces in weakly general position) Let f be a meromorphic mapping of Cm into Pn (C). Let Qi (i = 1, . . . , q) be slow (with respect to f ) moving hypersurfaces of Pn (C) in weakly general position with deg Qi = di . Put d be the least common multiple of d1 , . . . , dn+2 , i. n+d e., d = lcm(d1 , . . . , dn+2 ), and N = − 1. n (i) If Qi (f ) 6≡ 0 (1 ≤ i ≤ q) and q ≥ nN + n + 1, then we have q

k

X 1 [N ] q (r) + o(Tf (r)). N Tf (r) ≤ nN + n + 1 d Qi (f ) i=1 i

˜ {Q }q and q ≥ N + 2, then we (ii) If f is algebraically nondegenerate over K j j=1 have q X q 1 [N ] k (r) + o(Tf (r)). N Tf (r) ≤ N +2 d Qi (f ) i=1 i 2.7. Two lemmas.

Lemma 2.1. Let f : Cm → Pn (C) be a meromorphic mapping which is nondegenerate over Id (V ), and {Qj }qj=1 be a family of hypersurfaces of Pn (C) in N −subgeneral position with respect to V, with degree deg Qj = dj (1 ≤ j ≤ q). Let d be the least common multiple of d1 , . . . , dq , that is d = lcm(d1 , . . . , dq ). Then    (2N − k + 1)HV (d) HV (d) − 1 1 1 k q− (1 − δf (Q1 )) − − k+1 d1 m2 + 1 m1 + 1  q X 1 HV (d) − 1 (1 − δf (Qj )) Tf (r) − d mj + 1 j=1 j ≤

  q X 1 HV (d) − 1 [H (d)−1] NQj V(f ),≤mj (r) + o(Tf (r)), 1− d m + 1 2 j=1 j

where m1 ≥ m2 ≥ · · · ≥ mq are integers or ∞.

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Proof. By the Second Main Theorem (2.1), we have 

k ≤ ≤ ≤ ≤ ≤

 (2N − k + 1)HV (d) q− Tf (r) k+1 q X 1 [HV (d)−1] (r) + o(Tf (r)) NQj (f ) d j=1 j

q i X 1 h [HV (d)−1] [H (d)−1] NQj (f ),≤mj (r) + NQj V(f ),>mj (r) + o(Tf (r)) d j=1 j   q X HV (d) − 1 1 [HV (d)−1] NQj (f ),≤mj (r) + NQj (f ),>mj (r) + o(Tf (r)) d mj + 1 j=1 j  q  X 1 HV (d) − 1  [HV (d)−1] [HV (d)−1] NQj (f ) (r) − NQj (f ),≤mj (r) + o(Tf (r)) NQj (f ),≤mj (r) + d mj + 1 j=1 j  q  X 1 HV (d) − 1  [H (d)−1] [H (d)−1] NQj V(f ),≤mj (r) + NQj (f ) (r) − NQj V(f ),≤mj (r) + o(Tf (r)). d mj + 1 j=1 j

Noting that NQj (f ) (r) ≤ (1 − δf (Qj ))Tf (r), we get from the above inequality that k

 (2N − k + 1)HV (d) Tf (r) k+1   q X 1 HV (d) − 1 [H (d)−1] NQj V(f ),≤mj (r) 1− ≤ d m + 1 j j=1 j 

q−

+

q X 1 HV (d) − 1 (1 − δf (Qj ))Tf (r) + o(Tf (r)), d mj + 1 j=1 j

thus,  (2N − k + 1)HV (d) Tf (r) q− k+1     q X 1 HV (d) − 1 HV (d) − 1 1 [H (d)−1] [H (d)−1] NQ1V(f ),≤m1 (r) + NQj V(f ),≤mj (r) 1− 1− d1 m1 + 1 d m + 1 j 2 j=2



k ≤

q X 1 HV (d) − 1 (1 − δf (Qj ))Tf (r) + o(Tf (r)) d mj + 1 j=1 j   HV (d) − 1 1 1 [H (d)−1] NQ1V(f ),≤m1 (r) − d1 m2 + 1 m1 + 1   q X HV (d) − 1 1 [H (d)−1] NQj V(f ),≤mj (r) 1− + d m + 1 2 j=1 j

+ ≤

+

q X 1 HV (d) − 1 (1 − δf (Qj ))Tf (r) + o(Tf (r)). d mj + 1 j=1 j

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[H (d)−1]

Together with NQ1V(f ),≤m1 (r) ≤ (1 − δf (Q1 ))Tf (r), the above inequality implies   (2N − k + 1)HV (d) Tf (r) k q− k+1   1 1 HV (d) − 1 (1 − δf (Q1 ))Tf (r). − ≤ d1 m2 + 1 m1 + 1   q X 1 HV (d) − 1 [H (d)−1] + NQj V(f ),≤mj (r) 1− d m + 1 j 2 j=1 +

q X 1 HV (d) − 1 (1 − δf (Qj ))Tf (r) + o(Tf (r)). d mj + 1 j=1 j

Hence, the lemma is proved.



By a similar discussion as in proof of Lemma 2.1 using Theorem 2.2 instead of Theorem 2.1, one can obtain the following lemma. Lemma 2.2. Let f be a meromorphic mapping of Cm into Pn (C). Let Qi (i = 1, . . . , q) be slow (with respect to f ) moving hypersurfaces of Pn (C) in weakly general position with deg Qi = di . Put d be the least common multiple of d1 , . . . , dn+2 , i. n+d e., d = lcm(d1 , . . . , dn+2 ), and N = − 1. n Then    N 1 1 q (1 − δf (Q1 )) − − k p d1 m2 + 1 m1 + 1  q X N 1 (1 − δf (Qj )) Tf (r) − d mj + 1 j=1 j ≤

  q X 1 N [N ] NQj (f ),≤mj (r) + o(Tf (r)), 1− d m + 1 j 2 j=1

where  nN + n + 1, if Qj (f ) 6≡ 0 and q ≥ nN + n + 1; p := ˜ {Q }q and q ≥ N + 2, N + 2, if f is algebraically nondegenerate over K j j=1 and m1 ≥ m2 ≥ · · · ≥ mq are integers or ∞. 3. Proof of Theorem 1.7 Suppose that f and g have reduced representations respectively as follows: f = (f0 : f1 : · · · : fn )

and g = (g0 : g1 : · · · : gn ).

Assume that f 6= g. Then there two distinct indices s and t in {0, 1, . . . , n} such that H := fs gt − ft gs 6≡ 0. By the assumptions (a) and (b) of the theorem, we get that any z in ∪qj=1 {z ∈ Cm : 0 < νQj (f ) (z) ≤ mj }

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must be a zero of H, and further more, νH (z) ≥

q X

min{νQj (f ),≤mj (z), 1}

j=1

outside an analytic subset of codimension at least two, which implies that NH (r) ≥

q X

[1]

NQj (f ),≤mj (r).

j=1

On the other hand, on can get from the definition of the characteristic function and Jensen formula that Z log |fs gt − ft gs |σm NH (r) = Sm (r) Z Z ≤ log ||f ||σm + log ||g||σm Sm (r)

Sm (r)

= Tf (r) + Tg (r).

Therefore, we obtain from two inequalities above that q X j=1

[1]

NQj (f ),≤mj (r) ≤ Tf (r) + Tg (r).

Similar discussion for g we have q X j=1

[1]

NQj (g),≤mj (r) ≤ Tf (r) + Tg (r).

Hence, 2(Tf (r) + Tg (r))

≥ =

q X

[1]

NQj (f ),≤mj (r) +

j=1

N

[1] d d Qj j

(r) + (f ),≤mj

[1]

NQj (g),≤mj (r)

j=1 q X

j=1

q X

q X

j=1

N

[1] d d

(r).

Qj j (g),≤mj

d d

By Lemma 2.1 for f and the hypersurfaces Qj j of the common degree d, we have    1 1 (2N − k + 1)HV (d) HV (d) − 1 − − k q− k+1 d m2 + 1 m1 + 1  q X 1 HV (d) − 1  − Tf (r) d mj + 1 j=1 ≤

  q X HV (d) − 1 1 [H (d)−1] 1− N dV (r) + o(Tf (r)), d d m + 1 2 Q j (f ),≤m j=1 j

j

UNIQUENESS PROBLEMS OF MEROMORPHIC MAPPINGS

11

which means    d(2N − k + 1)HV (d) 1 1 k qd − − (HV (d) − 1) − k+1 m2 + 1 m1 + 1  q X HV (d) − 1  Tf (r) − mj + 1 j=1  q  X HV (d) − 1 [H (d)−1] 1− N dV (r) + o(Tf (r)) d m + 1 2 Qj j (f ),≤mj j=1  q  HV (d) − 1 X [1] N d (r) + o(Tf (r)). ≤ (HV (d) − 1) 1 − dj m2 + 1 j=1 Q (f ),≤m

≤

j

j

Similarly, for g we also have    1 1 d(2N − k + 1)HV (d) − (HV (d) − 1) − k qd − k+1 m2 + 1 m1 + 1  q X HV (d) − 1  − Tg (r) mj + 1 j=1  q  HV (d) − 1 X [1] N d (r) + o(Tg (r)). ≤ (HV (d) − 1) 1 − dj m2 + 1 j=1 Q (g),≤m j

j

Henceforth, we get form the inequalities above that    1 1 d(2N − k + 1)HV (d) − (HV (d) − 1) − k qd − k+1 m2 + 1 m1 + 1  q X HV (d) − 1  (Tf (r) + Tg (r)) − m + 1 j j=1   HV (d) − 1 ≤ 2(HV (d) − 1) 1 − (Tf (r) + Tg (r)) + o(Tf (r) + Tg (r)), m2 + 1 which implies q X

k

j=3

≤

mj (Tf (r) + Tg (r)) mj + 1

  d(2N − k + 1)HV (d) qd 4 − 2HV (d) q+ − − ) (Tf (r) + Tg (r)) (k + 1)(HV (d) − 1) HV (d) − 1 m2 + 1 +o((Tf (r) + Tg (r))).

This is a contradiction. Therefore, we completely prove the theorem. 4. Proof of Theorem 1.8 The proof of this theorem is some similar as the proof of Theorem 1.7. Assume that f and g have reduced representations respectively as follows: f = (f0 : f1 : · · · : fn )

and g = (g0 : g1 : · · · : gn ).

12

T. B. CAO AND H. Z. CAO d d

Then by Lemma 2.2 for f and the hypersurfaces Qj j of the common degree d, we have     X q 1 N q 1 N 1  Tf (r)  − k − − p d m2 + 1 m1 + 1 d m + 1 j j=1   q X 1 N [N ] 1− N d (r) + o(Tf (r)), d d m + 1 2 Q j (f ),≤m j=1

≤

j

j

which means that 

 qd − N p

k

 1−

≤



1 1 − m2 + 1 m1 + 1

N m2 + 1

X q



−

[N ]

N

q X j=1

 N  Tf (r) mj + 1

(r) + o(Tf (r)).

d d

Qj j (f ),≤mj

j=1

Similarly for g, 

 qd − N p

k

 1−

≤



1 1 − m2 + 1 m1 + 1

N m2 + 1

X q

N

j=1



−

[N ] d d

q X j=1

 N  Tg (r) mj + 1

(r) + o(Tg (r)).

Qj j (g),≤mj

Hence,   X q N  1 1 − (Tf (r) + Tg (r)) − m2 + 1 m1 + 1 mj + 1 j=1    X q N [N ] N [Nd] 1− (r) + N d (r) + o(Tf (r) + Tg (r)), dj dj m2 + 1 j=1 Q (f ),≤m Q (g),≤m



 qd − N p

k ≤



j

j

j

j

where  nN + n + 1, if Qj (f ) 6≡ 0 and q ≥ nN + n + 1; p := ˜ {Q }q and q ≥ N + 2, N + 2, if f is algebraically nondegenerate over K j j=1 Assume that f 6= g. Then there two distinct indices s and t in {0, 1, . . . , n} such that H := fs gt − ft gs 6≡ 0.

Set S = ∪{∩kj=0 νQij (f ) ; 1 ≤ i0 < . . . < ik ≤ q}. Then S is either an analytic subset of codimension at least two of Cm or an empty set. By the assumptions (a) and (b) of the theorem, we get that any z in ∪qj=1 {z ∈ Cm : 0 < νQj (f ) (z) ≤ mj } \ S

must be a zero of H, and further more, νH (z) ≥

q 1X min{νQj (f ),≤mj (z), 1} k j=1

UNIQUENESS PROBLEMS OF MEROMORPHIC MAPPINGS

13

outside the analytic subset S of codimension at least two, which implies that NH (r) ≥

q 1 X [1] (r). N k j=1 Qj (f ),≤mj

On the other hand, we can get also from the definition of the characteristic function and Jensen formula that Z log |fs gt − ft gs |σm NH (r) = Sm (r) Z Z log ||g||σm log ||f ||σm + ≤ Sm (r)

Sm (r)

= Tf (r) + Tg (r).

Therefore, we obtain from two inequalities above that q 1 X [1] (r) ≤ Tf (r) + Tg (r). N k j=1 Qj (f ),≤mj

Similar discussion for g we have

Hence, we get that

q 1 X [1] (r) ≤ Tf (r) + Tg (r). N k j=1 Qj (g),≤mj

2(Tf (r) + Tg (r))

≥ =

≥

q q 1 X [1] 1 X [1] (r) NQj (f ),≤mj (r) + N k j=1 k j=1 Qj (g),≤mj

q q 1 X [1] 1 X [1] N d N d (r) + (r) k j=1 Q dj (f ),≤m k j=1 Q dj (g),≤m j j j j   q 1 X  [N ] [N ] N d (r) + N d (r) d d kN j=1 Q j (f ),≤m Q j (g),≤m j

j

j

j

Henceforth, we get that    X  q N qd 1 1  −N  (Tf (r) + Tg (r)) k − − p m2 + 1 m1 + 1 m + 1 j j=1   N ≤ 2kN 1 − (Tf (r) + Tg (r)) + o(Tf (r) + Tg (r)), m2 + 1

which implies

q X

k

j=3

≤

mj (Tf (r) + Tg (r)) mj + 1

  qd 2kN − 2 q + 2k − 2 − − ) (Tf (r) + Tg (r)) pN m2 + 1 +o((Tf (r) + Tg (r))).

This is a contradiction. Therefore, the theorem is completely proved.

14

T. B. CAO AND H. Z. CAO

References [1] Y. Aihara, Unicity theorems for meromorphic mappings with deficiencies, Complex Variables 42(2000), 259-268. [2] T. B. Cao, K. Liu, H. Z. Cao, Multiple value and uniqueness problem for meromorphic mappings sharing hyperplanes, Ann. Ponlon. Math. 107 (2013), 153-166. [3] T. B. Cao and H. X. Yi, Uniqueness theorems for meromorphic mappings sharing hyperplanes in general position, Sci. Sin. Math. 41(2011), No. 2, 135-144.(in Chinese) [4] Z. H. Chen and Q. M. Yan, Uniqueness theorem of meromorphic mappings into PN (C) sharing 2N + 3 hyperplanes regardless of multiplicities, Int. J. Math. 20, 717(2009), 717726. [5] G. Dethloff and T. V. Tan, A uniqueness theorem for meromorphic maps with moving hypersurfaces, Publ. Math. Debrecen 78(2011), 347–357. [6] M. Dulock, M. Ru, A uniqueness theorem for holomorphic curves sharing hypersurfaces, Complex Variables and Elliptic Equations 53(2008), No. 8, 797-802. [7] H. Fujimoto, The uniqueness promble of meromorphic maps into the complex projective spaces, Nagoya Math. J. 58(1975), 1-23. [8] P. C. Hu, P. Li and C. C. Yang, Unicity of Meromorphic Mappings, Kluwer 2003. [9] F. L¨ u,The uniqueness problem for meromorphic mappings with truncated multiplicities, Kodai Math. J. 35(2012), 485-499. [10] R. Nevanlinna, Eindentig keitss¨ atze in der theorie der meromorphen funktionen, Acta. Math. 48(1926), 367-391. [11] H. T. Phuong, Uniqueness theorems for holomorphic curves sharing moving hypersurfaces, Complex Variables and Elliptic Equations 2012, 1-11, iFirst. [12] H. T. Phuong, On unique range sets for holomorphic maps sharing hypersurfaces without counting multiplicity, Acta. Math. Vietnamica 34(2009), No. 3, 351-360. [13] S. D. Quang, Unicity of meromorphic mappings sharing few hyperplanes, Ann. Polon. Math. 102(2011), No. 3, 255-270. [14] S. D. Quang, Second main theorem and unicity of meromorphic mappings for hypersurfaces of projective varieties in subgeneral position, arXiv: 1302.1261v1 [math. CV] 6 Feb 2013 [15] S. D. Quang, Second main theorem and uniqueness problem of meromorphic mappings with moving hypersurfaces, arXiv: 1302.0412v1 [math. CV] 2 Feb 2013 [16] L. Smiley, Geometric conditions for unicity of holomorphic curves, Contemp. Math. 25(1983), 149-154. [17] Z. H. Tu, Z. H. Wang, Uniqueness problem for meromorphic mappings in several complex variables into Pn (C) with truncated multiplicities, Acta. Math. Sci. 33B(1)(2013), 122-130. [18] L. Yang, Multiple values of meromorphic functions and functions combination, Acta Math. Sin. Chinese Ser. 14(1964), 428-437.(in Chinese) [19] H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press 1995/Kluwer 2003. (Ting-Bin Cao) Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China E-mail address: [email protected](Corresponding author) (Hong-Zhe Cao) Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China E-mail address: [email protected]