Algebraic dependences of meromorphic mappings sharing moving hyperplanes without counting multiplicities

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0028 Mathematical and Physical Sci., 2017, Vol. 62, Iss. 8, pp. 23-31 This paper is available online at ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS SHARING MOVING HYPERPLANES WITHOUT COUNTING MULTIPLICITIES Ha Huong Giang Faculty of Fundamental Sciences, Electric Power University Abstract. In this article, we will prove a algebraic dependences theorem for meromorphic mappings sharing moving hyperplanes with different multiple and a general condition on the intersections of the inverse images of these hyperplanes. Keywords: Algebraic dependence, meromorphic mapping, truncated, multiplicity. 1. Introduction In 1989, W. Stoll [1] studied the theory on algebraic dependences of meromorphic mappings in several complex variables into the complex projective spaces for fixed targets. In 2011, M. Ru [2] generalized W. Stoll’s result to the case of holomorphic curves into the complex projective spaces sharing moving hyperplanes. Recently, by using the new second main theorem given by Thai-Quang [3], P. D. Thoan, P. V. Duc and S. D. Quang in [4], [5] and [6] gave some improvements of the results of W. Stoll and M. Ru. In order to state some of their result, we first recall the following. A meromorphic mapping of Cm into Pn(C)∗ be called moving hyperplane in Pn(C). Let a1, ..., aq (q ≥ n + 1) be q moving hyperplanes with reduced representations aj = (aj0 : ... : ajn) (1 ≤ j ≤ q). We say that a1, ..., aq are in general position if det(ajkl) 6≡ 0 for any 1 ≤ j0 < j1 < ... < jn ≤ q. Let fi : Cm → Pn(C) (1 ≤ i ≤ λ) be meromorphic mappings with reduced representations fi := (fi0 : ... : fin). Let aj (1 ≤ j ≤ q) be moving hyperplanes in Pn(C) in general position with reduced representations aj := (aj0 : ... : ajn). Put (fi, aj) :=∑n s=0 fisajs 6= 0 for each 1 ≤ i ≤ λ, 1 ≤ j ≤ q and assume that min{1, ν0(f1,aj),≤kj} = ... = min{1, ν0(fλ,aj),≤kj}. In, P. D. Thoan, P. V. Duc and S. D. Quang [5] proved an better algebraic dependences theorem as follows. Received September 11, 2017. Accepted September 30, 2017. Contact Ha Huong Giang, e-mail: hhgiang79@yahoo.com. 23 Ha Huong Giang Theorem A. Let f1, ..., fλ : Cm → Pn(C) be non-constant meromorphic mappings. Let ai (1 ≤ i ≤ q) be moving hyperplanes in Pn(C) in general position such that T (r, ai) = o(max1≤i≤λT (r, fj)) (1 ≤ i ≤ q) and (fi, aj) 6≡ 0 for 1 ≤ i ≤ λ, 1 ≤ j ≤ q. Assume that the following conditions are satisfied. (a) min{1, ν(f1,aj)} = ... = min{1, ν(fλ,aj)} for each 1 ≤ j ≤ q, (b) dim{z| (f1, ai)(z) = (f1, aj)(z) = 0} ≤ m− 2 for each 1 ≤ i < j ≤ q, (c) there exists an integer number l, 2 ≤ l ≤ λ, such that for any increasing sequence 1 ≤ j1 < ... < jl ≤ λ, fj1(z) ∧ ... ∧ fjl(z) = 0 for every point z ∈ ⋃q i=1(f1, gi) −1{0}. If q > n(2n+1)λ−(n−1)(λ−1)λ−l+1 , then f1 ∧ ... ∧ fλ ≡ 0. The above results are the best results on the algebraic dependences of meromorphic mappings sharing moving hyperplanes available at the present. Actually, there are many authors consider the multiple values for meromorphic mappings sharing hyperplanes, i.e., consider only the intersecting points of the mappings fi and the hyperplanes aj with the multiplicity not exceed a certain number kj < +∞. In 2016, L. N. Quynh [7] generalized and improved Theorems A by considering the multiple values problem and reducing the number of hyperplanes. Moreover, the condition (b) is very important role in above authors’ proof. In 2012, H. H. Giang, L. N. Quynh and S. D. Quang [8] introduced new techniques to treat the case where the condition (b) is replaced by a more general. Our purpose in this paper is to generalize Quynh’s result by considering the general case where codim of the intersections of inverse images of arbitrary d, (d ≥ 1) hyperplanes be at least two . Namely, we will prove the following. Theorem 1.1. Let f1, ..., fλ : Cm → Pn(C) be non-constant meromorphic mappings. Let {aj}qj=1 be q moving hyperplanes in Pn(C) in general position satisfying T (r, aj) = o(max1≤i≤λ T (r, fi)) (1 ≤ j ≤ q). Let kj (1 ≤ j ≤ q) be positive integers or +∞. Assume that (fi, aj) 6≡ 0 for 1 ≤ i ≤ λ, 1 ≤ j ≤ q, and the following conditions are satisfied. (a) min{1, ν0(f1,aj),≤kj} = ... = min{1, ν0(fλ,aj),≤kj} for each 1 ≤ j ≤ q, (b) dim (⋂d+1 i=1 f −1 1 (aji) ) ≤ m− 2 for each 1 ≤ j1 < ... < jd+1 ≤ q, (c) there exists an integer number l, 2 ≤ l ≤ λ, such that for any increasing sequence 1 ≤ i1 < ... < il ≤ λ, fi1(z) ∧ ... ∧ fil(z) = 0, for every point z ∈ ⋃q j=1 f −1 1 (aj). We assume further that rankR{aj}f1 = ... = rankR{aj}fλ = N + 1, whereN is a positive integer. If q∑ j=1 1 kj + 1−N < q N(2n−N + 2) − λdq q(d(λ− l) + 1) + λ(N − 1) . then f1 ∧ ... ∧ fλ ≡ 0. 24 Algebraic dependences of meromorphic mappings sharing moving hyperplanes... 2. Some second main theorems and lemmas 2.1. We set ||z|| = (|z1|2 + · · ·+ |zm|2)1/2 for z = (z1, . . . , zm) ∈ Cm and define B(r) := {z ∈ Cm : ||z|| < r}, S(r) := {z ∈ Cm : ||z|| = r} (0 < r < +∞). Define vn−1(z) := (ddc||z||2)n−1 and σn(z) := d c log ||z||2 ∧ (ddc log ||z||2)n−1 on Cm\0. 2.2. For a divisor ν on Cm, we denote byN(r, ν) the counting function of the divisor ν as usual in Nevanlinna theory (see [9]). For a positive integerM orM =∞, we define the truncated divisors of ν by ν[M ](z) = min{M,ν(z)}, ν[M ]≤k (z) := { ν[M ](z) if ν[M ](z) ≤ k 0 if ν[M ](z) > k. Similarly, we define ν[M ]>k . We will write N [M ](r, ν), N [M ] ≤k (r, ν), N [M ] >k (r, ν) for N(r, ν [M ]), N(r, ν [M ] ≤k ), N(r, ν [M ] >k ) as respectively. Let ϕ : Cm −→ C be a meromorphic function. Define Nϕ(r) = N(r, νϕ), N [M ] ϕ (r) = N [M ](r, νϕ), N [M ] ϕ,≤k(r) = N [M ] ≤k (r, νϕ), N [M ] ϕ,>k(r) = N [M ] >k (r, νϕ). We will omit the character [M ] ifM =∞. 2.3. Let f : Cm → Pn(C) be a meromorphic mapping. For arbitrarily fixed homogeneous coordinates (w0 : ... : wn) on Pn(C), we take a reduced representation f = (f0 : ... : fn), which means that each fi is a holomorphic function on Cm and f(z) = (f0(z) : ... : fn(z)) outside the analytic set I(f) = {f0 = ... = fn = 0} of codimension ≥ 2. Let a be a moving hyperplane in Pn(C) with reduced representation a = (a0 : ... : an). We denote by T (r, f) the characteristic function of f and bymf,a(r) the proximity function of f with respect to a (see [10]). If (f, a) 6≡= 0, then the first main theorem for moving targets in value distribution theory states T (r, f) + T (r, a) = mf,a(r) +N(f,a)(r). Theorem 2.4. (The First Main Theorem for general position [1]). Let fj : Cm → Pn(C), (1 ≤ j ≤ λ) be meromorphic mappings located in general position. Assume that 1 ≤ k ≤ n. Then N(r, µf1∧...∧fλ) +m(r, f1 ∧ ... ∧ fλ) ≤ ∑ 1≤i≤λ T (r, fi) +O(1). Here, by µf1∧...∧fλ we denote the divisor associated with f1 ∧ ... ∧ fλ. We also denote by Nf1∧...∧fλ(r) the counting function associated with the divisor µf1∧...∧fλ . Let V be a complex vector space of dimension n ≥ 1. The vectors {v1, ..., vk} are said to be in general position if for each selection of integers 1 ≤ i1 < ... < ip ≤ k with p ≤ n, then 25 Ha Huong Giang vi1 ∧ ... ∧ vip 6= 0. The vectors {v1, ..., vk} are said to be in special position if they are not in general position. Take 1 ≤ p ≤ k, then {v1, ..., vk} are said to be in p-special position if for each selection of integers 1 ≤ i1 < ... < ip ≤ k, the vectors {vi1 , ..., vip} are in special position. Theorem 2.5. (The SecondMain Theorem for general position [1]). LetM be a connected complex manifold of dimension m. Let A be a pure (m-1)-dimensional analytic subset of M . Let V be a complex vector space of dimension n+ 1 > 1. Let p and k be integers with 1 ≤ p ≤ k ≤ n+ 1. Let fj : M → P(V ), 1 ≤ j ≤ k, be meromorphic mappings. Assume that f1, ..., fk are in general position. Also assume that f1, ..., fk are in p-special position on A. Then we have µf1∧...∧fk ≥ (k − p+ 1)νA. Here by νA we denote the reduced divisor whose support is the set A. The following is a new second main theorem given by S. D. Quang [10], which is an improvement the second main theorem of Thai-Quang in [3]. Theorem 2.6.(The Second Main Theorem for moving target [10]). Let f : Cm → Pn(C) be a meromorphic mapping. Let {ai}qi=1 (q = 2n−N+2) be moving hyperplanes in Pn(C) in general position such that (f, ai) 6≡ 0 (1 ≤ i ≤ q), where N + 1 = rankR{ai}(f). Then we have || q 2n−N + 2Tf (r) ≤ q∑ i=1 N [N ] (f,ai) (r) + o(Tf (r)) +O(max1≤i≤qTai(r)). As usual, by the notation ” || P " we mean the assertion P holds for all r ∈ [0,∞) excluding a Borel subset E of the interval [0,∞) with ∫E dr <∞. Remark: With the assumption of Theorem 2.6, we see that || Tf (r) ≤ N(2n−N + 2) q q∑ i=1 N [1] (f,ai) (r) + o(Tf (r)) +O( max 1≤i≤q Tai(r)) ≤ n(n+ 2) q q∑ i=1 N [1] (f,ai) (r) + o(Tf (r)) +O( max 1≤i≤q Tai(r)). (2..1) 3. Proof of Theorem 1.1. It suffices to prove Theorem 1.1 in the case of λ ≤ n+ 1. We set N = rankR{aj}(fi)− 1. Proposition 3.1.[see [7]] For every 1 ≤ i ≤ λ, 1 ≤ j ≤ q and 1 ≤ N ≤ n, we have N [N ] (fi,aj),≤kj (r) ≥ kj + 1 kj + 1−NN [N ] (fi,aj) (r)− N kj + 1−N T (r, fi). Proposition 3.2.[see [7]] Let hi : Cm → Pn(C) (1 ≤ i ≤ p ≤ n + 1) be meromorphic mappings with reduced representations hi := (hi0 : ... : hin). Let ai (1 ≤ i ≤ n + 1) be moving hyperplanes in Pn(C) with reduced representations ai := (ai0 : ... : ain). Put h˜i := ((hi, a1) : ... : (hi, an+1)). Assume that a1, ..., an+1 are located in general position such that (hi, aj) 6≡ 0 (1 ≤ i ≤ p, 1 ≤ j ≤ n+ 1). Let S be a pure (m− 1)-dimensional analytic subset of 26 Algebraic dependences of meromorphic mappings sharing moving hyperplanes... Cm such that S 6⊂ (a1∧...∧an+1)−1{0}. Then h1∧...∧hp = 0 on S if and only if h˜1∧...∧h˜p = 0 on S. Suppose that f1 ∧ ... ∧ fλ 6≡ 0. For λ indices 0 = j0 < j1 < ... < jλ−1 ≤ n such that (f1, aj0) ... (fλ, aj0) (f1, aj1) ... (fλ, aj1) ... ... ... (f1, ajλ−1) ... (fλ, ajλ−1)  is nondegenerate. Put J = {j0, ..., jλ−1}, Jc = {1, ..., q}\J and BJ =  (f1, aj0) ... (fλ, aj0) (f1, aj1) ... (fλ, aj1) ... ... ... (f1, ajλ−1) ... (fλ, ajλ−1)  Proposition 3.3. If BJ is nondegenerate, i.e., detBJ 6≡ 0 then 1 d ∑ j∈J (min1≤i≤λ{ν0(fi,aj),≤kj} −min{1, ν0(f1,aj),≤kj}) + 1 d q∑ j=1 (d(λ− l) + 1)min{1, ν0(f1,aj),≤kj} ≤ µf˜1∧...∧f˜λ , on the set Cm\ ( A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)) where f˜i := ((fi, aj0) : ... : (fi, ajλ−1)) and A = ⋃ 1≤i<j≤q Z(f1, ai) ∩ Z(f1, aj). Proof. Indeed, putA := ⋃j∈J Z(f1, aj), Ac := ⋃j∈Jc Z(f1, aj). We consider the following two cases. * Case 1. Let z0 ∈ A\ ( A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)) be a regular point of A. Then z0 is a zero of one of the meromorphic functions {(f1, aj)}j∈J . Without loss of generality we may assume that z0 is a zero of (f1, aj0). Let S be an irreducible component of A containing z0. Suppose that U is an open neighborhood of z0 in Cm such that U ∩ {A\S} = ∅. Choose a holomorphic function h on an open neighborhood U ⊂ U of z0 such that νh(z) = 1 d min1≤i≤λ{ν0(fi,aj0 ),≤kj0 (z)} if z ∈ S and νh(z) = 0 if z 6∈ S. Then (fi, aj0) = aih (1 ≤ i ≤ λ), where ai are holomorphic functions. Therefore, the matrix  (f1, aj1) ... (fλ, aj1)... ... ... (f1, ajλ−1) ... (fλ, ajλ−1)  is of rank ≤ λ− 1. Hence, there exist λ holomorphic functions b1, ..., bλ, not all zeros, such that λ∑ i=1 bi.(fi, ajk) = 0 (1 ≤ k ≤ λ− 1). 27 Ha Huong Giang Without loss of generality, we may assume that the set of common zeros of {bi}λi=1 is an analytic subset of codimension ≥ 2. Then there exists an index i1, 1 ≤ i1 ≤ λ, such that S 6⊂ b−1i1 {0}. We may assume that i1 = λ. Then for each z ∈ (U ′ ∪ S)\b−1λ {0}, we have f˜1(z) ∧ ... ∧ f˜λ(z) = f˜1(z) ∧ ... ∧ f˜λ−1(z) ∧ ( f˜λ(z) + λ−1∑ i=1 bi bλ f˜i(z) ) = f˜1(z) ∧ ... ∧ f˜λ−1(z) ∧ (V (z)h(z)) = h(z)∆(f˜1(z) ∧ ... ∧ f˜λ−1(z) ∧ V (z)), where V (z) := (aλ + ∑λ−1 i=1 bi bλ ai, 0, ..., 0). By the assumption and by the Proposition 3.2, for any increasing sequence 1 ≤ i1 < ... < il ≤ λ − 1, we have f˜i1 ∧ ... ∧ f˜il = 0 on S. This implies that the family {f˜1, ..., f˜λ−1, V } is in (l + 1)-special position on S. By using The Second Main Theorem for general position [1], we have µf˜1∧...∧f˜λ−1∧V (z) ≥ λ− l, ∀ z ∈ S. Hence µf˜1∧...∧f˜λ−1(z) ≥ νh(z) + λ− l = 1 d min1≤i≤λ{ν0(fi,aj0 ),≤kj0 (z)}+ λ− l, for all z ∈ (U ′ ∪ S)\b−1i1 {0}. This implies that 1 d ∑ j∈J (min1≤i≤λ{ν0(fi,aj),≤kj (z0)} −min{1, ν0(f1,aj),≤kj (z0)}) + 1 d q∑ j=1 (d(λ− l) + 1)min{1, ν0(f1,aj),≤kj (z0)} = 1 d min1≤i≤λ{ν0(fi,aj0 ),≤kj0 (z0)}+ λ− l ≤ µf˜1∧...∧f˜λ(z0). * Case 2. Let z0 ∈ Ac\ ( A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)) be a regular point of Ac. Then z0 is a zero of (f1, aj), j ∈ Jc. By the assumption and by the Proposition 3.2, the family {f˜1, ..., f˜λ} is in l-special position on each irreducible component of Ac containing z0. By using The Second Main Theorem for general position [1], we have µf˜1∧...∧f˜λ(z0) ≥ λ− l + 1. Hence 1 d ∑ j∈J (min1≤i≤λ{ν0(fi,aj),≤kj (z0)} −min{1, ν0(f1,aj),≤kj (z0)}) + 1 d q∑ j=1 (d(λ− l) + 1)min{1, ν0(f1,aj),≤kj (z0)} = λ− l + 1 ≤ µf˜1∧...∧f˜λ(z0). 28 Algebraic dependences of meromorphic mappings sharing moving hyperplanes... From the above two cases we get the desired inequality of the Proposition. Proof Theorem 1.1. For each j, 1 ≤ j ≤ q, we set Nj(r) = λ∑ i=1 N [N ] (fi,aj),≤kj (r)− ((λ− 1)N + 1)N [1] (f1,aj),≤kj (r). For each permutation I = (j1, ..., jq) of (1, ..., q), we set TI = {r ∈ [1,+∞);Nj1(r) ≥ ... ≥ Njq(r)}. It is clear that ⋃ I TI = [1,+∞). Therefore, there exists a permutation, for instance it is I0 = (1, ..., q), such that ∫ TI0 dr = +∞. Then we have N1(r) ≥ N2(r) ≥ ... ≥ Nq(r), for all r ∈ TI0 . By the assumption for f1∧ ...∧ fλ 6≡ 0, there exist indices J = {j0, ..., jλ−1} with 0 = j0 < j1 < ... < jλ−1 ≤ n such that detBJ 6≡ 0. We note that N1(r) = Nj0(r) ≥ Nj1(r) ≥ ... ≥ Njλ−1(r) ≥ Nn(r), for each r ∈ TI0 . We see that min1≤i≤λai ≥ ∑λ i=1min{N, ai} − (λ − 1)N for every λ non-negative integers a1, ..., aλ. Then the Proposition 3.3 implies that 1 d ∑ j∈J ( λ∑ i=1 min{N, ν0(fi,aj),≤kj} − ((λ− 1)N + 1)min{1, ν0(f1,aj),≤kj} ) + 1 d q∑ j=1 (d(λ− l) + 1)min{1, ν0(f1,aj),≤kj} ≤ µf˜1∧...∧f˜λ , on the set Cm\ ( A ∪⋃λi=1 I(fi) ∪ (aj0 ∧ ... ∧ ajλ−1)−1(0)). Integrating both sides of this inequality, we have 1 d ∑ j∈J ( λ∑ i=1 N [N ] (fi,aj),≤kj (r)− ((λ− 1)N + 1)N [1] (f1,aj),≤kj (r) ) + 1 d q∑ j=1 (d(λ− l) + 1)N [1](f1,aj),≤kj (r) ≤ Nf˜1∧...∧f˜λ(r) = NdetBJ (r). (3..1) Also, by Jensens formula, we have NdetBJ (r) ≤ ∫ S(r) log |detBJ |σn +O(1) ≤ λ∑ i=1 T (r, fi) + o( max 1≤i≤λ T (r, fi)). (3..2) 29 Ha Huong Giang Set T (r) = ∑λ i=1 T (r, fi). Combining (3..1) and (3..2), then for all r ∈ I0, we have || T (r) ≥ 1 d λ−1∑ i=0 Nji(r) + q∑ j=1 (λ− l + 1 d )N [1] (f1,aj),≤kj (r) + o(T (r)) ≥ λ dq q∑ j=1 Nj(r) + q∑ j=1 (λ− l + 1 d )N [1] (f1,aj),≤kj (r) + o(T (r)) = q∑ j=1 ( λ− l + 1 d − λ((λ− 1)N + 1) dq ) N [1] (f1,aj),≤kj (r) + q∑ j=1 λ dq λ∑ i=1 N [N ] (fi,aj),≤kj (r) + o(T (r)) ≥ λ∑ i=1 q∑ j=1 ( λ dq + d(λ− l) + 1 dλN − (λ− 1)N + 1 Ndq ) N [N ] (fi,aj),≤kj (r) + o(T (r)) ≥ λ∑ i=1 q∑ j=1 q(d(λ− l) + 1) + λ(N − 1) λNdq ( N [N ] (fi,aj) (r)− N kj + 1−N T (r, fi) ) + o(T (r)) ≥ q(d(λ− l) + 1) + λ(N − 1) λNdq  λ∑ i=1 q∑ j=1 N [N ] (fi,aj) (r)− q∑ j=1 N kj + 1−N T (r)  + o(T (r)) ≥ q(d(λ− l) + 1) + λ(N − 1) λNdq  q 2n−N + 2 − q∑ j=1 N kj + 1−N T (r) + o(T (r)). Letting r → +∞, we have 1 ≥ q(d(λ− l) + 1) + λ(N − 1) λNdq  q 2n−N + 2 − q∑ j=1 N kj + 1−N  . Thus q∑ j=1 1 kj + 1−N ≥ q N(2n−N + 2) − λdq q(d(λ− l) + 1) + λ(N − 1) . This is a contradiction. Thus, we have f1 ∧ ... ∧ fλ = 0. The theorem is proved. 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