MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION —————————– Pham Duc Thoan ON THE RELATION DEFECT AND THE ALGEBRAIC DEPENDENCES OF MEROMORPHIC MAPPINGS Specialized: Geometry and Topology Code: 62.46.10.01 SUMMARY DOCTOR OF PHILOSOPHY IN MATHEMATICS Hanoi, 01-2011 Thesis was completed at: Hanoi national University of Education Science instructor: Prof Dr Do Duc Thai Rewier 1: Prof Dr Nguyen Van Mau, Hanoi University of ScienceVietnam National University Rewier 2: Prof Dr Le Hung Son, Hanoi University of Technology Rewier 3: Prof Dr Nguyen Van Khue, Hanoi National University of Education Thesis will be approved by School committee at hour date month year Thesis can be found at: -Viet Nam national library -Library of Hanoi National University of Education Introduction Reasons for selecting topics In the late 20’s last century, Nevanlinna foundated the value distribution theory of the meromorphic function of a variable Over the next decade many mathematicians in the world such as H Cartan, W Stoll, PA Griffiths, L Carlson, P Vojta, J Noguchi interest in research and develop on Nevanlinna theory for more general object class So far, Nevanlinna theory has become one of the most important theory of mathematics with many beautiful theorems have been proved The most striking result is that the inequality in terms of defects and unicity theorem By the attractive nature of the geometric theory, we have chosen the theme ”On the relation defect and the algebraic dependences of meromorphic mappings” Specifically, we focus on research and has given some results on the defects for meromorphic functions to P1(C) and meromorphic mappings to Pn(C), and we also study the algebraic dependences and apply these results to the study unicity problem with truncated multiplicity for meromorphic mappings of several complex variables The aim and subject of thesis The main aim of the thesis is to study the meromorphic having maximal defect sum and the algebraic dependence of the meromorphic mappings Stadying subject is the meromorphic mappings with a maximum with maximal defect sum and the algebraic dependence of the meromorphic mappings Studying methods used in thesis Using knowledges about Complex Geomatry and Complex Analysis, Nevanlinna theory Simultaneously, we also created new techniques to solve the issues raised in the thesis The first is the study of the maximal defect sum of the meromorphic function, we have devised a ”noise” by ”small” function The second is the study of unique problems of meromorphic mapping, the authors often proved directly and through the second fundamental theorem Here, we approach the problem with the theory of ”algebraic dependence” of the meromorphic mappings of several complex variables that W Stoll proposed The results of the thesis Among the theorems that Nevanlinna proved, the theorem about the relationship of defect to keep a special role Namely, the theorem is stated the following: Theorem A If f be a nonconstant meromorphic function on C then δ(a, f ) a∈P1 (C) Theorem A was proved for the class meromorphic functions of complex variables For example, theorem Cartan-Nochka said that if f : C → Pn(C) be a linearly nondependence holomorphic function and {Hj }q−1 be hyperplanes in N -subgeneral position in Pn(C) j=0 then q−1 [n] i=0 δ (Hi , f ) ≤ 2N − n + Have a question naturally arises: What we can say about the class of functions f which is the maximal defect sum? In other words, we can say about the equal sign in inequality occurring defects? This problem were many mathematicians research interest in recent times For example, N Toda proved the following theorem: Theorem B Let f : Cm −→ Pn(C) be a linearly nondegenerate, and let {Hj }q j=1 be hyperplanes in N -subgeneral position in Pn(C), where ≤ n < N and 2N − n + < q ≤ +∞ Assume that δ(Hj , f ) > (1 ≤ j ≤ q) and q δ [n] (Hj , f ) = 2N − n + j=1 Then one of the following two statements holds: 2N − n + + of the hyperplanes Hj n+1 at which f has deficiency value 1, i.e δ(Hj , f ) = 1, (I) There are at least (II) {Hj }q has a Borel distribution j=1 Continue the above research , in the first two chapters of the thesis we study the class meromorphic mappings which has a maximal defect sum Namely, in Chapter we showed the necessary condition for the class meromorphic function has a maximal defect function, also indicate that the meromorphic function is very small Specifically, we have proved the following two theorems Theorem 1.3.1 Let f : C → P1(C) be a meromorphic function of finite order For each n ≥ 1, define gn(z) = f (z n), ∀z ∈ C and hn(z) = f n(z), ∀z ∈ C Then we have necessarily λ := ρf ∈ Z+ and λ equals the lower order of f 1) If there exists n0 ≥ such that a∈C δ(a, gn0 ) = 2) If there exists a sequence {ni}+∞ ⊂ Z+ such that i=1 2, ∀i ≥ a∈C δ(a, hni ) = Theorem 1.3.2 Let f : Cm → P1(C) be a meromorphic function of finite order satisfying λ := ρf ∈ Z and / a∈C δ(a, f ) = Denote by A the set of all nonconstant meromorphic functions h : Cm → P1(C) such that Th(r) = o Tf (r) , TDh (r) = o TDf (r) Then, for each h ∈ A, we have a∈C δ(a, f + h) − 2k(λ) < 2, where k(λ) is a positive constant which depends only on λ Chapter of the thesis has extended the results of N Toda for class meromorphic mappings of several variables that have maximal defect sum for moving targets Namely, we have proved the following theorem Theorem 2.3.1 Let f : Cm −→ Pn(C) be a nonconstant meromorphic mapping, and let {ai}q−1 be ”small” (with respect i=0 to f ) meromorphic mappings of Cm into Pn(C) in N − subgeneral position such that f is linearly nondegenerate over R({ai}q−1), i=0 where ≤ n < N and 2N − n + < q < +∞ Suppose further that f has nonzero deficiency value at for each ≤ i ≤ q − and q−1 j=0 δ (aj , f ) = 2N − n + Then one of the following two statements holds 2N − n + (I) There are at least [ ] + of the moving targets aj n+1 at which f has deficiency value 1, i.e δ(aj , f ) = , q−1 (II) n is odd and the family {aj }j=0 has a Borel distribution In 1926, Nevanlinna proved that if f and g be two non-constant meromorphic functions on C such that f −1(ai) = g −1(ai) at 5 distinction point a1, · · · , a5 then f ≡ g In the context of the review theorem point of Nevanlinna for meromorphic function of several complex variables into complex projective space, in 1975 H Fujimoto proved the following important theorem Theorem C Let Hi (1 ≤ i ≤ 3N + 2) be 3N + hyperplanes in general position in PN (C), f and g be two non-constant meromorphic mappings from Cn to PN (C) such that f (Cn) Hi, g(Cn) Hi Assume that v(f,Hi) = v(g,Hi) with ≤ i ≤ 3N +2 Then, if f or g be linearly nondependence then f ≡ g In the last decade many works have continued to develop on the results of H Fujimoto and has formed a research direction in the Nevanlinna theory is the study problem unicity (also known as the unicity theorem.) In particular, the unicity theorem has been studied continuously in recent years and has obtained deep results Among the approaches to the unicity problem has a method by W Stoll proposed, research unicity problem that is through study of the algebraic dependence of meromorphic mappings Development of the above ideas of W Stoll, in 2001 M.Ru showed unicity theorem for holomorphic curves into complex projective space with moving targets Namely, M Ru proved the following: Theorem D Let f and g be non-constant meromorphic functions if there exists distinction meromorphic functions a1, a2, · · · , a7 such that Taj (r) = o(max{Tf (r), Tg (r)}) (0 ≤ j ≤ 7) and f (z) = aj (z) ⇔ g(z) = aj (z) then f ≡ g Continue the above research , in the chapters of the thesis, we have showed some unicity theorem for meromorphic mappings of several variables to complex projective space through the study of the algebraic dependence of their mapping The results that we achieved a significant expansion for the theorems of M Ru Namely, we proved the following theorem: Theorem 3.2.4 Let f1, · · · , fλ : Cm → Pn(C) be nonconstant meromorphic mappings Let gi : Cm → Pn(C) (1 ≤ i ≤ q) be moving targets located in general position such that T (r, gi) = o(max1≤j≤λ T (r, fj )) (1 ≤ i ≤ q) and (fi, gj ) ≡ for ≤ i ≤ λ, ≤ j ≤ q Let κ be a positive integer or κ = +∞ and κ = min{κ, n} Assume that the following conditions are satisfied i) min{κ, v(f1,gj )} = · · · = min{κ, v(fλ,gj )} for each ≤ j ≤ q, ii) dim{z|(f1, gi)(z) = (f1, gj )(z) = 0} ≤ m − for each ≤ i < j ≤ q, iii) There exists an integer number l, ≤ l ≤ λ, such that for any increasing sequence ≤ j1 < · · · < jl ≤ λ, fj1 (z) ∧ · · · ∧ fjl (z) = for every point z ∈ ∪q (f1, gi)−1{0} i=1 Then n(2n + 1)λ − (κ − 1)(λ − 1) , then f1, · · · , fλ are λ−l+1 algebraically dependent over C, i.e f1 ∧ · · · ∧ fλ ≡ on Cm i) If q > ii) If fi, ≤ i ≤ λ are linearly nondegenerate over R {gj }q j=1 and n(n + 2)λ − (κ − 1)(λ − 1) , λ−l+1 then f1, · · · , fλ are algebraically dependent over C q> iii) If fi, ≤ i ≤ λ are linearly nondegenerate over C, gi (1 ≤ i ≤ q) are constant mappings and (q − n − 1)((λ − 1)(κ − 1) + q(λ − l + 1)) ≤ qnλ, then f1, · · · , fλ are algebraically dependent over Cm Theorem 3.3.1 Let f1, f2 : Cm → Pn(C) be non-constant meromorphic mappings Let gj : Cm → Pn(C) be moving targets located in general position and T (r, gj ) = o(max1≤i≤2{T (r, fi)}), ≤ j ≤ q and (fi, gj ) ≡ for ≤ i ≤ 2, ≤ j ≤ q Let κ be a positive integer or κ = +∞ and κ = min{κ, n} Assume that the following conditions are satisfied i) min{κ, v(f1,gj )(z)} = min{κ, v(f2,gj )} for each z ∈ Cm and 1≤j≤q ii) dim{(f1, gi)−1{0} ∩ (f1, gj )−1(z)} ≤ n − for each ≤ i < j≤q iii) f1(z) = f2(z) for each z ∈ ∪q (f1, gj )−1{0} j=1 If q > 2n(2n + 1) − 2(κ − 1) then f1 ≡ f2 Structure of the thesis The layout of the thesis out of the introduction and the conclusion, the thesis consists of three chapters Chapter I: ”On meromorphic functions with maximal defect sum” Chapter II: ”Meromorphic mappings with maximal defect sum for moving targets” Chapter III: ”The algebraic dependence of the meromorphic mappings and applications” Chapter On meromorphic functions with maximal defect sum we show the necessary conditions of the maximality of defect sum and show that the class of meromorphic functions with maximal defect sum is very thin in the sense that deformations of meromorphic functions with maximal defect sum by small meromorphic functions are not meromorphic functions with maximal defect sum This chapter is based on the article [4] Theorem of classical Nevanlinna deficiency relation was pointed out that if f : Cm −→ P1(C) be a meromorphic function then a∈P1 (C) δ(a, f ) A question naturally arises: What can be said about the class of meromorphic functions f is a∈P1 (C) δ(a, f ) = 2? This problem has research interests of many mathematicians, such as N Toda, J Lu and Y Yasheng As stated in the introduction, the purpose of this chapter is to continue to study the problem on the meromorphic function of the fixed target Namely, we show the necessary conditions of the maximality of defect sum Later, we show that the class of meromorphic functions with maximal defect sum is very thin in the sense that deformations of meromorphic functions with maximal defect sum by small meromorphic functions are not meromorphic functions with maximal defect sum Further more, we can measure the deviation of defect sum of meromorphic functions before and after by constant 11 hn(z) = f n(z), ∀z ∈ C Then we have necessarily λ := ρf ∈ Z+ and λ equals the lower order of f 1) If there exists n0 ≥ such that a∈C δ(a, gn0 ) = 2) If there exists a sequence {ni}+∞ ⊂ Z+ such that i=1 a∈C δ(a, hni ) 2, ∀i ≥ Theorem 1.3.2 Let f : Cm → P1(C) be a meromorphic function of finite order satisfying λ := ρf ∈ Z and / δ(a, f ) = a∈C Denote by A the set of all nonconstant meromorphic functions h : Cm → P1(C) such that Th(r) = o Tf (r) , TDh (r) = o TDf (r) Then, for each h ∈ A, we have a∈C δ(a, f + h) − 2k(λ) < 2, where k(λ) is a positive constant which depends only on λ = 12 Chapter Meromorphic mapping with maximal with defect sum for moving targets This chapter is based on the article [3] The chapter for the study of the meromorphic mappings from Cm to Pn(C) with maximal defect sum for moving targets For about 20 years, the study of Nevanlinna theory for moving targets were more interested in mathematics This is a natural extension when we replace the hyperplane (or hypersurface) fixed in the complex projective space with a small coefficient of the function One of the most important results of this research is the theorem Cartan-Nochka for moving targets proved by M Ru and W Stoll Theorem Let f : Cm −→ Pn(C) be a nonconstant meromorq−1 phic mapping and assume that {ai}i=0 be ”small” meromorphic mappings corresponding to f from Cm to Pn(C) in N-subgeneral position such that f is linearly nondependence on R({ai}q−1) i=0 Then q−1 j=0 δ (aj , f ) ≤ 2N − n + Thus there is a question naturally arises:What we can say about the functions f which is the number of defects for maximal? In other words, we can extend the results of N Toda for the meromorphic mappings of several variables that have maximal defect sum with moving targets or not? The main purpose of this chapter is to answer that question 2.2 The initial results 13 First we recall two of the Nochka weight Lemma for moving targets How to prove they are repeated all the claims corresponding to the fixed hyperplane Lemma 2.2.1 Let {ai}i∈Q be q moving targets in Pn(C) in N subgeneral position, and assume that q > 2N − n + Then there are positive rational constants ωj , j ∈ Q satisfying the following: i) < ωj ≤ 1, ∀j ∈ Q, ˜ ii) Setting ω = maxj∈Q ωj , one gets 1) + n + n n+1 ≤ω≤ ˜ iii) 2N − n + N iv) For R ⊂ Q with < |R| ≤ N + 1, q j=1 ωj ˜ = ω (q − 2N + n − j∈R ωj ≤ rank{ai}i∈R ˜ The above ωj are called N ochka weights, and ω the N ochka constant We will denote θ = ω −1 for later convenience ˜ Lemma 2.2.2 Let q > 2N − n + 1, and let {ai}i∈Q be q moving targets in PnC in N -subgeneral position Let {ωj }j∈Q be its Nochka weights Let Ej ≥ 1, j ∈ Q be arbitrarily given numbers Then for every subset R ⊂ Q with < |R| ≤ N + 1, there is a subset Ro ⊂ R such that |Ro| = rank{ai}i∈R and ωi i∈R Ei ≤ i∈Ro Ei The purpose of this section is to prove lemma plays an important role in proving the theorem on the deficiency of meromorphic mappings with maximal deficiency sum for moving targets Lemma 2.2.7 Let f be a meromorphic mapping of Cm into Pn(C) with a reduced representation f = (f0 : · · · : fn) Consider N > n and q be any integer satisfying 2N −n+1 < q < +∞ Put 14 Q = {0, , q − 1} Let X = {aj : j ∈ Q} be the set of ”small” (with respect to f ) meromorphic mappings from Cm into Pn(C) in N -subgeneral position Assume that f is nondegenerate over q−1 R({ai}i=0 ) and any function ω : Q → (0, 1] satisfy the condition (iv) in Lemma 2.2.1, we have q−1 j=0 ω(j) 2.3 · δ (aj , f ) ≤ n + The meromorphic mapping with maximal defi- ciency sum In this section, we use the above lemmas to prove the theorems on the deficiency of meromorphic mappings with maximal deficiency sum for moving targets Theorem 2.3.1 Let f : Cm −→ Pn(C) be a nonconstant q−1 meromorphic mapping, and let {ai}i=0 be ”small” (with respect to f ) meromorphic mappings of Cm into Pn(C) in N − subgeneral position such that f is linearly nondegenerate over R({ai}q−1), i=0 where ≤ n < N and 2N − n + < q < +∞ Suppose further that f has nonzero deficiency value at for each ≤ i ≤ q − and q−1 j=0 δ (aj , f ) = 2N − n + Then one of the following two statements holds 2N − n + ] + of the moving targets aj at n+1 which f has deficiency value 1, i.e δ(aj , f ) = , (I) There are at least [ (II) n is odd and the family {aj }q−1 has a Borel distribution j=0 15 Chapter The algebraic dependences of meromorphic mappings and applications This chapter for the study of algebraic dependence of the meromorphic mapping from Cm to Pn(C) with moving targets in general position and application to the study of unicity problem The chapter is based on the article [1] end [2] The theory on algebraic dependences of meromorphic mappings in several complex variables into the complex projective spaces for fixed targets is studied by W Stoll in 1989 Later, Min Ru generalized Stoll’s result to holomorphic curves into the complex projective spaces for moving targets and show some unicity theorems of holomorphic curves into the complex projective spaces for moving targets As far as we know, they are the first results on the unicity problem for moving targets We now state his remarkable results Let g1, , gq (q ≥ n) be q meromorphic mappings of Cm into Pn(C) with reduced representations gj = (gj0 : · · · : gjn) (1 ≤ j ≤ q) We say that g1, , gq are located in general position if det(gjk l ) ≡ for any ≤ j0 < j1 < < jN ≤ q Let Mn be the field of all meromorphic functions on Cm Denote q by R gj j=1 ⊂ Mn the smallest subfield which contains C and gjk with gjl ≡ all gjl Let f be a meromorphic mapping of Cm into Pn(C) with reduced representation f = (f0 : · · · : fn) We say that f is linearly nondegenerate over R gj q j=1 if f0, , fN are linearly independent 16 over R gj Let ft : C q j=1 m → Pn(C) (1 t λ) be meromorphic mappings with reduced representations ft := (ft0 : · · · : ftn) Let gj : Cm → Pn(C) (1 j q) be moving targets located in general position with reduced representations gj := (gj0 : · · · : gjN ) Assume that (ft, gj ) := N i=0 fti gji = for each ≤ t ≤ λ, ≤ j ≤ q and (f1, gj )−1{0} = · · · = (fλ, gj )−1{0} Put Aj = (f1, gj )−1{0} for each j q Assume that every analytic set Aj has the irriducible t j decomposition as follows Aj = ∪i=1Aji(1 tj +∞) Set A = ∪Aji≡Akl {Aji ∩ Akl } with ≤ i ≤ tj , ≤ l ≤ tk , ≤ j, k ≤ q Denote by T [n+1, q] the set of all injective maps from {1, · · · n+1} to {1, · · · , q} For each z ∈ Cm \ {∪β∈T [n+1,q]{z|gβ(1)(z) ∧ · · · ∧ gβ(n+1)(z) = 0} ∪ A ∪ ∪λ I(fi)}, we define ρ(z) = {j|z ∈ Aj } i=1 Then ρ(z) ≤ n For any positive number r > 0, define ρ(r) = sup{ρ(z)||z| ≤ r}, where the supremum is taken over all z ∈ Cm \ {∪β∈T [n+1,q]{z|gβ(1)(z) ∧ · · · ∧ gβ(n+1)(z) = 0} ∪ A ∪ ∪λ I(fi)} i=1 Then ρ(r) is a decreasing function Let d := lim ρ(r) r→+∞ Then d ≤ n If for each i = j, dim{Ai ∩ Aj } ≤ m − 2, then d = We state the following M Ru’ theorems Theorem A Let f1, · · · , fλ : C → Pn(C) be non-constant holomorphic curves Let gi : C → Pn(C) (1 ≤ i ≤ q) be moving targets located in general position and T (r, gi) = o(max1≤j≤λ T (r, fj )) (1 ≤ i ≤ q) Assume that (fi, gj ) ≡ for ≤ i ≤ λ, ≤ j ≤ q and Aj := (f1, gj )−1{0} = · · · = (fλ, gj )−1{0} for each ≤ j ≤ q Denote A = ∪q Aj Let j=1 17 l, ≤ l ≤ λ, be an integer such that for any increasing sequence ≤ j1 < · · · < jl ≤ λ, fj1 (z) ∧ · · · ∧ fjl (z) = for every point dn2(2n + 1)λ , then f1, · · · , fλ are algebraically z ∈ A If q > λ−l+1 over C, i.e f1 ∧ · · · ∧ fλ ≡ on C Theorem B In addition to the assumption in Theorem A we assume further that fi, ≤ i ≤ λ, are linearly nondegenerated Then f1, · · · , fλ are algebraically dependent over C, i.e f1 ∧· · ·∧ dn(n + 2)λ fλ ≡ on C, if q > λ−l+1 Use the second main theorems of Do Duc Thai and Si Duc Quang, in the fisrt part of chapter we extend to the result of M Ru decreased significantly by moving targets The unicity theorems with truncated multiplicities of meromorphic mappings of Cm into the complex projective space Pn(C) sharing a finite set of fixed (or moving) hyperplanes in Pn(C) have received much attention in the last few decades We now state a recent result of Do Duc Thai and Si Duc Quang which is the one of the best results available at present Theorem C Let f : Cm → Pn(C) be a meromorphic mapping Let κ be a positive integer Let {aj }q be ”small” (with respect j=1 to f ) meromorphic mappings of Cm into Pn(C) in general position such that dim{z ∈ Cm : (f, ai)(z) = (f, aj )(z) = 0} ≤ m−2 (1 ≤ i < j ≤ q) Assume that f is linearly nondegen- erate over R({aj }q ) We denote F(f, {aj }q , κ) be the set of j=1 j=1 all linearly nondegenerate over R({aj }q ) meromorphic maps j=1 g : Cm → Pn(C) satisfying the conditions: (i) (v(f,aj ), κ) = (v(g,aj ), κ) (1 ≤ j ≤ q), 18 (ii) f (z) = g(z) on q j=1 {z ∈ Cn : (f, aj )(z) = 0} Then we have F(f, {aj }q , 1) = 1, j=1 (i) If q = 2n2 + 4n and n ≥ 2, then where denote by S the cardinality of the set S (3n + 1)(n + 2) (ii) If q = and n ≥ 2, then F(f, {aj }q , 2) ≤ j=1 In the above-mentioned theorems, there is an strong assumption on the nondegeneracy of meromorphic mappings over R({aj }q ) j=1 Thus, naturally arises the study of unicity theorems without this assumption Inspired of the argument of Do Duc Thai anh Si Duc Quang, Z Chen, Y Li and Q Yan showed successfully the following unicity theorem of such kind without that assumption Theorem D f : Cm → Pn(C) be a nonconstant meromorphic mapping Let κ be a positive integer Let {aj }q be ”small” j=1 (with respect to f ) meromorphic mappings of Cm into Pn(C) in general position such that (f, aj ) ≡ (1 ≤ j ≤ q) and dim{z ∈ Cm : (f, ai)(z) = (f, aj )(z) = 0} ≤ m − (1 ≤ i < j ≤ q) We denote G(f, {aj }q , κ) be the set of all meromorphic maps j=1 g : Cm → Pn(C) satisfying the conditions: (i) (v(f,aj ), κ) = (v(g,aj ), κ) (ii) f (z) = g(z) on q j=1 {z (1 ≤ j ≤ q), ∈ Cn : (f, aj )(z) = 0} Then If q = 4n2 + 2n and n ≥ 2, then G(f, {aj }q , 1) = j=1 How to say about the unicity theorems with truncated multiplicities in the case where q < 4n2 + 2n? It seems to us that some key techniques in their proofs could not be used when q < 4n2 + 2n In the last of chapter 3, we give the answer for the above problem 19 Our approach is based on the theorem ”algebraic dependences” of meromorphic mappings 3.1 The some auxiliary results The following, Do Duc Thai and Si Duc Quang have proved the Second Main Theorem for moving targets Theorem 3.1.2 Let f : Cm → Pn(C) be a meromorphic mapping Let {a1, , aq } (q = 2) be a set of q meromorphic mappings of Cm into Pn(C) located in general position such that f is linearly nondegenerate over R T (r, f ) ≤ [n] q i=1 N(f,ai ) (r) gj q j=1 q n+2 Then · + O max1≤i≤q T (r, ai) +o T (r, f ) Theorem 3.1.3 Let f : Cm → Pn(C) be a meromorphic mapping Let A = {a0, , aq−1} (q ≥ 2n + 1) be a set of q meromorphic mappings of Cm into Pn(C) located in general position such that (f, ai) ≡ for each ≤ i ≤ q Then T (r, f ) ≤ [n] q i=1 N(f,ai ) (r)+O q 2n+1 · max1≤i≤q T (r, ai) +O log+ T (r, f ) Theorem 3.1.4 [The First Main Theorem for general position] Let fi : Cm → Pn(C), (1 ≤ i ≤ λ) be meromorphic mappings located in general position Assume that ≤ λ ≤ n+1 Then N (r, µf1∧···∧fλ ) + m(r, f1 ∧ · · · ∧ fλ) ≤ 1≤i≤λ T (r, fi ) + O(1) Theorem 3.1.5 [The Second Main Theorem for general position] Let M 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 + > Let p and k be integers with ≤ p ≤ k ≤ n + Let fj : M → P (V ), ≤ 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)vA 20 3.2 Algebraic dependences of meromorphic mappings With the same assumption on the nondegeneracy of small moving targets, it is our main purpose of this section to show some algebraic dependence theorems of meromorphic mappings from Cm into Pn(C) for moving targets in more general situations Theorem 3.2.1 Let f1, · · · , fλ : Cm → Pn(C) be nonconstant meromorphic mappings Let gi : Cm → Pn(C) (1 ≤ i ≤ q) be moving targets located in general position and T (r, gi) = o(max1≤j≤λ T (r, fj )) (1 ≤ i ≤ q) Assume that (fi, gj ) ≡ for ≤ i ≤ λ, ≤ j ≤ q and Aj := (f1, gj )−1{0} = · · · = (fλ, gj )−1{0} for each ≤ j ≤ q Denote A = ∪q Aj Let j=1 l, ≤ l ≤ λ, be an integer such that for any increasing sequence ≤ j1 < · · · < jl ≤ λ, fj1 (z) ∧ · · · ∧ fjl (z) = for every point z ∈ A Then f1, · · · , fλ are algebraically over C, i.e dn(2n + 1)λ f1 ∧ · · · ∧ fλ ≡ on Cm, if q > λ−l+1 Theorem 3.2.3 In addition to the assumption in Theorem 3.2.1 we assume further that fi, ≤ i ≤ λ, are linearly nondegenerate over R {gj }q Then f1, · · · , fλ are algebraically dej=1 dn(n + 2)λ pendent over C, i.e f1 ∧ · · · ∧ fλ ≡ on Cm, if q > λ−l+1 Theorem 3.2.4 Let f1, · · · , fλ : Cm → Pn(C) be nonconstant meromorphic mappings Let gi : Cm → Pn(C) (1 ≤ i ≤ q) be moving targets located in general position such that T (r, gi) = o(max1≤j≤λ T (r, fj )) (1 ≤ i ≤ q) and (fi, gj ) ≡ for ≤ i ≤ λ, ≤ j ≤ q Let κ be a positive integer or κ = +∞ and κ = min{κ, n} Assume that the following conditions are satisfied 21 i) min{κ, v(f1,gj )} = · · · = min{κ, v(fλ,gj )} for each ≤ j ≤ q, ii) dim{z|(f1, gi)(z) = (f1, gj )(z) = 0} ≤ m − for each ≤ i < j ≤ q, iii) There exists an integer number l, ≤ l ≤ λ, such that for any increasing sequence ≤ j1 < · · · < jl ≤ λ, fj1 (z) ∧ · · · ∧ fjl (z) = for every point z ∈ ∪q (f1, gi)−1{0} i=1 Then n(2n + 1)λ − (κ − 1)(λ − 1) , then f1, · · · , fλ are λ−l+1 algebraically dependent over C, i.e f1 ∧ · · · ∧ fλ ≡ on Cm i) If q > ii) If fi, ≤ i ≤ λ are linearly nondegenerate over R {gj }q j=1 and n(n + 2)λ − (κ − 1)(λ − 1) , λ−l+1 then f1, · · · , fλ are algebraically dependent over C q> iii) If fi, ≤ i ≤ λ are linearly nondegenerate over C, gi (1 ≤ i ≤ q) are constant mappings and (q − n − 1)((λ − 1)(κ − 1) + q(λ − l + 1)) ≤ qnλ, then f1, · · · , fλ are algebraically dependent over C 3.3 The unicity theorem with truncated multiplicities of meromorphic mappings In this section, we study the unicity problem with truncated multiplicities of meromorphic mappings in several complex variables into the complex projective spaces by algebraic dependences Theorem 3.3.1 Let f1, f2 : Cm → Pn(C) be non-constant meromorphic mappings Let gj : Cm → Pn(C) be moving targets located in general position and T (r, gj ) = o(max1≤i≤2{T (r, fi)}), ≤ j ≤ q and (fi, gj ) ≡ for ≤ i ≤ 2, ≤ j ≤ q Let κ be a pos- 22 itive integer or κ = +∞ and κ = min{κ, n} Assume that the following conditions are satisfied i) min{κ, v(f1,gj )(z)} = min{κ, v(f2,gj )} for each z ∈ Cm and 1≤j≤q ii) dim{(f1, gi)−1{0} ∩ (f1, gj )−1(z)} ≤ m − for each ≤ i < j≤q iii) f1(z) = f2(z) for each z ∈ ∪q (f1, gj )−1{0} j=1 If q > 2n(2n + 1) − 2(κ − 1) then f1 ≡ f2 23 Conclusion and recommentdations Conclusions The main results of the thesis • Pointed out some necessary condition for meromorphic function of a class of maximal deficiency and showed that the class of meromorphic function with maximal deficiency sum is very ”thin” in the sense that if the ”noise” of the meromorphic function of the maximal sum meromorphic by the meromorphic function of ”small”, they no longer are meromorphic function of the maximal deficiency sum of shortcomings as well • Pointed out some theorem about mappings of several complex variable with maximal deficiency sum for moving targets • Proved three theorems about algebraic dependence of the meromorphic mapping from Cm to Pn(C) with moving targets in general position • Proved the unicity theorem with truncated multiplicities of meromorphic mappings in several complex variables with targets of q < 4n2 + 2n in situation no assumption about the non degenerate linear mapping of the meromorphic of f : Cm → Pn(C) Recommendations on further research In Chapter 1, we have only studied the of the defect of meromorphic function from Cm to P1(C) Naturally, should make the theorem is similar to theorems 1.3.1 and theorem 1.3.2 for meromorphic mappings from Cm to complex projective space Pn(C) with n ≥ 24 In chapter 2, theorem 2.3.1 on the maximal deficiency sum of meromorphic mapping for moving targets may be multiple blocks Can we improve the proof of the theorem in Chapter to reduce the number of moving targets or not? With questions and analysis, further research by our as follows: To study the class meromorphic mappings into the high dimension complex projective space with maximal deficiency sum for moving targets and fixed targets taking into account the multiplier block Improving the second main theorem and counting multiples to reduce the number of moving targets can cause a degenerate for the meromorphic mapping Continue research on the unicity problem only when there is progress on the second main theorem and theorems about algebraic degeneracy Due to limited time we can not get the results to the problems posed We hope that this problem will soon be resolved in the near future 25 List of publicated works related to the thesis [1] Pham Duc Thoan, Pham Viet Duc and Si Duc Quang, (2010) A unicity theorem with truncated multiplicities of meromorphic mappings in several complex variables, Submitted to Bull Math de la Soc des Sciences Math de Roumanie [2] Duc Thoan Pham and Viet Duc Pham, (2010) Algebraic dependences of meromorphic mappings in several complex variables, Ukrainian Math J 62, No 7, p 923-936 [3] Duc Thoan Pham and Viet Duc Pham, On meromorphic mappings in several complex variables with maximal deficiency sum for moving targets, c nhn xt tt ca phn bin v gi bn sa cha ti Acta Math Vietnamica [4] Pham Duc Thoan and Le Thanh Tung, (2011) On meromorphic functions with maximal defect sum, Ann Polon Math 100, No 2, p 115-125 ... study of unicity theorems without this assumption Inspired of the argument of Do Duc Thai anh Si Duc Quang, Z Chen, Y Li and Q Yan showed successfully the following unicity theorem of such kind... received much attention in the last few decades We now state a recent result of Do Duc Thai and Si Duc Quang which is the one of the best results available at present Theorem C Let f : Cm → Pn(C) be... · ·∧ dn(n + 2)λ fλ ≡ on C, if q > λ−l+1 Use the second main theorems of Do Duc Thai and Si Duc Quang, in the fisrt part of chapter we extend to the result of M Ru decreased significantly by moving